Geometry 10

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Find the image of O(0, 0) after two reflections, first in y = 4, and then in x = -7.

(-14, 8)

What is the image of the point (-3, 4) after a rotation of 90° counterclockwise about the point (-3, 0)?

(-7, 0)

Find the center and radius of (x + 8)2 + (y + 4)2 = 49.

(-8, -4); 7

If a point P(-2, 1) is reflected across the line x = -1, what are the coordinates of its reflection image?

(0, 1)

If a point P(1, 3) is reflected across the line y = 2, what are the coordinates of its reflection image?

(1, 1)

If a point P(1, -2) is reflected across the line x = 3, what are the coordinates of its reflection image?

(5, -2)

What is the image of the point (0, 9) after a rotation 90° clockwise about the origin?

(9, 0)

diagonal

(geometry) a straight line connecting any two vertices of a polygon that are not adjacent

deductive reasoning

(sometimes called logical reasoning) is the process of reasoning logically from given statements or facts to a conclusion. Pg. 113

Write the standard equation for the circle with center (-16, 30) that passes through (0, 0).

(x + 16)2 + (y - 30)2 = 1156

Write the standard equation for the circle with center (-2, 5) that passes through (1, 9). (x - 2)2 + (y + 5)2 = 5 (x + 2)2 + (y - 5)2 = 5 (x + 2)2 + (y - 5)2 = 25 (x - 2)2 + (y + 5)2 = 25

(x + 2)2 + (y - 5)2 = 25

A low-watt radio station can be heard only within a certain distance from the station. On the graph below, the circular region represents that part of the city where the station can be heard, and the center of the circle represents the location of the station. Which equation represents the boundary for the region where the station can be heard?

(x + 5)2 + (y + 4)2 = 25

Addition Property

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Alternate interior angles

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Applying the Triangle Sum Theorem

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Biconditional

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CPCTC what it is and how to apply

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Corresponding Angles

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Deductive Reasoning

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Division Property

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Exterior Angles

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How to use the Properties of Iscosceles and Equilateral Triangles

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Identifying Congruent Parts, Congruent Triangles and Writing Congruent Statements

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Identifying the types of Triangles

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Inductive Reasoning

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Interior Angles

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Laws of Detachment

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Laws of Logic

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Laws of Syllogism

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Multiplication Property

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Perpendicular Lines

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Reflexive Property

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Ruler Postulate

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Same side interior angles

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Segment Addition Postulate

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Substitution Property

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Subtraction Property

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Symmetric Property

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Transitive Property

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Writing Proofs Using SSS, SAS, ASA, AAS, and HL plus CPCTC

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acute angle

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adjacent angles

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angle

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angle bisector

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area

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axiom

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collinear points

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compass

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complementary angles

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congruent angles

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construction

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coplanar

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distance formula

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intersect

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line

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midpoint formula

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net

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obtuse angle

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opposite rays

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parallel lines

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parallel planes

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perimeter

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perpendicular bisector

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perpendicular lines

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plane

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point

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postulate

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ray

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right angle

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segment

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skew lines

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space

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straight angle

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straightedge

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supplementary angles

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vertical angles

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Tickmarks are used to...

...indicate equal lengths in line segments

To find the midpoint,...

...take the average of the coordinates of the endpoints on a number line. (x+y)/2

0.32 0.36 0.5 0.26

0.32

A circle with a diameter of 2 inches and a square with 2-inch sides have the same center. Find the area of the region that is inside the square and outside the circle. Use 3.14 for π.

0.9 in.2

1 : 112 1 : 2304 2 : 4205 1 : 12,544

1 : 12,544

Cylinder A has a radius of 1 and a height of 4. Cylinder B has a radius of 2 and a height of 4. Find the ratio of the volumes of the two cylinders. (There is not supposed to be a picture for this problem - If you'd like, you may draw the cylinders to help find the ratio of the volumes.)

1 : 4

If the ratio of the radii of two spheres is 1 : 8, what is the ratio of the surface areas of the two spheres?

1 : 64

obtuse angle

1 obtuse triangle

right triangle

1 right angle

Ruler postulate

1) The point on a line can be paired with real numbers in such a way that any two points can have coordinates 0 and 1. 2) Once a coordinate system has been chosen, the distance between two points = the absolute value of the difference of their coordinates.

If one measurement of a golden rectangle is 8.2 inches, which could be the other measurement?

1.618 in.

The width of a golden rectangle is 3 m, which is shorter than the length. What is the length?

1.85 m

The Community Recreation Center is developing plans for a new sports facility. Community members can submit suggestions for the new facility, along with basic scale drawings of their ideas. Lupe wants to include a new 65- by 100-meter soccer field in the athletic center. She is submitting a scale drawing on an 8.5- by 11-inch sheet of paper. Which scale should Lupe use to create as large a drawing as possible on the paper?

1/16=1m

If a dart hits the target at random, what it the probability that it will land in the shaded region?

1/9

Find the geometric mean of 20 and 5.

10

In movies and television, the ratio of the width of the screen to the height is called the aspect ratio. Television screens usually have an aspect ratio of 4 : 3, while movie screens usually have an aspect ratio of 1.85 : 1. However, if a movie is made for television in "Letterbox" format, it retains the 1.85 : 1 aspect ratio and fills in the top and bottom parts of the screen with black bars. What would be the height of a movie in "Letterbox" format on a television screen that measures 25 inches along its diagonal? (Hint: First find the width and height of the television screen.)

10.81 in.

Find the length of side a to the nearest tenth.

10.9

Solve the problem below. 104° 121.5° 110° none of these

104°

Chef Imelda can do something unique. Using a secret process, she can bake a nearly perfectly spherical pie consisting of a chicken filling inside a thick crust. The radius of the whole pie is 19 cm, and the radius of the filling is 16 cm. What is the volume of the crust alone, to the nearest tenth of a unit? Use π ≈ 3.14.

11,567.8 cm3

If one measurement of a golden rectangle is 6.8 inches, which could be the other measurement?

11.002 in.

Find the geometric mean of 48 and 3.

12

dedecagon

12 sided

Find the area of a parallelogram with vertices at P(-8, -3), Q (-7, 3), R(-9, 3), and S(-10, -3).

12 square units

The volumes of two similar solids are 729 m3 and 64 m3. The surface area of the larger one is 648 m2. What is the surface area of the smaller one?

128 m2

15.6 14.3 23.1 9.1

14.3

15.8 14.6 19.7 13.6

14.6

A sphere has a volume of 288π ft3. Find the surface area of the sphere.

144π ft2

Find the surface area of the solid. Round to the nearest square foot. 147 ft2 245 ft2 196 ft2 98 ft2

147 ft2

15.8 14.6 19.7 13.6

15.8

Find the surface area of the cylinder to the nearest whole number. 79 m2 16 m2 25 m2 158 m2

158 m2

To find the height of a flagpole, a surveyor moves 90 feet away from the base of the flagpole and then, with a transit 3 feet tall, measures the angle of elevation to the top of the flagpole to be 60°. What is the height of the flagpole? Round your answer to the nearest foot.

159 ft

Find the surface area of a sphere that has a diameter of 4 cm.

16π cm2

Solve the question below. 14 15 16 17

17

Find the volume of the cone in terms of pi, and rounded to the nearest cubic meter. 70,583π m3; 221,744 m3 17,645π m3; 55,433 m3 52,937.5π m3; 166,308 m3 26,468.75π m3; 83,154 m3

17,645π m3; 55,433 m3

A blueprint for a house has a scale of 1 inch: 30 inches. A wall in the blueprint is 7 in. What is the length of the actual wall?

17.5 ft

A footbridge is in the shape of an arc of a circle. The bridge is 7 ft tall and 28 ft wide. What is the radius of the circle that contains the bridge? Round your answer to the nearest tenth.

17.5 ft

A regular nonagon has a radius of 18.8 cm. What is the length of the apothem? Round your answer to the nearest tenth.

17.7 cm

Find the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle.

17/13

Use formulas to find the surface area of the prism. Show your answer to the nearest hundredth.

170.16 cm2

Find the area of a regular octagon with perimeter 48 cm.

173.8 cm2

Interior angle sum

180 (n-2)

straight

180 degrees

In pottery class, Ludolf made a cylindrical vase that is 32 cm tall. Its base has a radius of 8 cm. He wants to paint the outside of the vase. How many square centimeters will Ludolf have to paint? Round to the nearest whole number. Use 3.14 as an approximation for pi.

1809 cm2

Find the volume of the prism. 942 m3 38 m3 945 m3 1890 m3

1890 m3

To find the height of a tall tree, a surveyor moves 140 feet away from the base of the tree and then, with a transit 4 feet tall, measures the angle of elevation to the top of the tree to be 53°. What is the height of the tree? Round your answer to the nearest foot.

190 ft

Linear pair

2 adj. angles that create a straight line

Linear pair

2 adjacent angles that make a straight line (are supplementary)

Definition of linear pair

2 angles are a linear pair if they are adjacent and non-common sides form opposite rays

definition of congruent angles

2 angles are congruent if they have equal measures

Vertical angles

2 angles that are accross from eachother when 2 lines intersect.

alternate interior angles (AIA)

2 angles that lie between the 2 lines and on the opp. side of the transversal.

Alternate Exterior Angles (AEA)

2 angles that like outside the 2 lines and on the opp. side of the transversal.

Adjacent Angles

2 angles that share a vertex and a side but no points in their interiors

Adjacent angles

2 angles that share a vertex and ray. They are next to each other (touching)

adjacent

2 angles that share the same vertex and share a side. Do not overlap eachother

Corresponding Angles

2 angles that sit in the same location at each intersection of the linear and the transversal

isosceles triangle

2 congruent sides

Plane

2 dimensional; looks like a top of a desk or a piece of paper; goes on in all directions (does not end)

definition of perpendicular lines

2 lines are perpendicular if they intersect to form a right angle

Collinear

2 or more points on the same line

Coplanar

2 or more points on the same plane

Opposite Rays

2 rays that have the same endpoint and go in opposite directions forming a line

Angle

2 rays that share a common endpoint

Opposite Rays

2 rays that share a common endpoint and go in opposite directions

opposite rays

2 rays with a shared endpoint going opposit directions, combined to make a line

definition of congruent segments

2 segments are congruent if they have equal measures

Find the value of x.

2 square root 21

Find the value of x, the slant height of the regular pyramid. Round your answer to the nearest hundredth, if necessary.

2.82 ft 4.47 ft 5.65 ft 6 ft

There is a law that the ratio of the width to length for the American flag should be 10 : 19. Which dimensions are in the correct ratio?

20 by 38 in.

A.23.2 B.20.9 C.6.9 D.6.2

20.9

Find the volume of the square pyramid. 18,468 ft3 6156 ft3 2052 ft3 3078 ft3

2052 ft3

If a dart hits the target at random, what it the probability that it will land in the unshaded region?

21/25

Julio had a job helping a jeweler. He had the assignment of counting the faces, vertices, and edges on the rubies. On the first ruby, Julio counted 30 vertices and 50 edges. He quickly realized he didn't have to count the faces. How many faces were there?

22 faces

Solve the problem below. 22° 112° 34° 68°

22°

Find the area of ΔABC. The figure is not drawn to scale.

23.14 cm2

Use a net to find the surface area of the prism.

240 m2

Find the volume of the cone. Use 3.14 for pi. 8000.72 in.3 2464.19 in.3 2476.41 in.3 190.49 in.3

2476.41 in.3

Find the surface area of the cone in terms of pi.

24π cm2

A machinist drilled a cone-shaped hole into a solid cube of metal as shown. If the cube's sides have a length of 7 centimeters, what is the volume of the metal cube after the cone is drilled? Use 3.14 for pi and round your answer to the nearest tenth.

253.2 cm3

Find the volume of the composite space figure. 70 cm3 238 cm3 196 cm3 266 cm3

266 cm3

PQ = 6 cm. How many points are in the locus of points that are 9 cm from point P and 3 cm from a line perpendicular to

3

Solve the question below. 3 4.5 6 7.5

3

acute triangle

3 acute angles

equiangular triangle

3 congruent angles

equalteral triangle

3 congruent sides

For our calculation, what number do we use for TT (pie)

3.14

What is the probability that a randomly dropped marker will fall in the shaded region?

3/4

Find the value of x to the nearest integer when tan x = 0.577.

30

Point O is the center of the circle. What is the value of x? (Assume that lines that appear to be tangent are tangent.) 12 60 15 30

30

Solve the problem below. 30° 150° 120° 60°

30°

2280 1320 3120 48

312

313 227 47 133

313

A large totem pole near Kalama, Washington, is 154 feet tall. On a particular day at noon it casts a 243-foot shadow. What is the sun's angle of elevation at that time?

32.4

A design on the surface of a balloon is 4 cm wide when the balloon holds 53 cm3 of air. How much air does the balloon hold when the design is 16 cm wide?

3392 cm3

If the ratio of the radii of two spheres is 6 : 5, what is the ratio of the surface areas of the two spheres?

36 : 25

Find the surface area of the solid. Round to the nearest square foot. 36 ft2 32 ft2 64 ft2 68 ft2

36 ft2

Angle of Rotation

360 divided by the number of rotational symmetries.

Solve the problem below. 76° 104° 142° 38°

38°

Find the circumference of the circle in terms of π.

38π in.

Determine AB. 4 or 9 4 1 1 or 6

4

Find the area of the circle. Use π = 3.14. 16.61060 m2 66.4424 m2 14.444 m2 4.12565 m2

4.12565 m2

Find the value of x, the slant height of the regular pyramid. Round your answer to the nearest hundredth, if necessary.

4.47 ft

Find the value of x, the slant height of the regular pyramid. Round your answer to the nearest hundredth, if necessary. 2.82 ft 4.47 ft 5.65 ft 6 ft

4.47 ft

For the pair of similar figures, give (a) the ratio of the perimeters and (b) the ratio of the areas of the first figure to the second one

4/3 16/9

Find the area of a parallelogram with vertices at A(-9, 5), B(-8, 10), C(0, 10), and D(-1, 5).

40

Find the surface area of a sphere that has a diameter of 20 cm. 1600π cm2 4000/3 pi cm2 100π cm2 400π cm2

400π cm2

Find the area of kite ABCD if BD = 48 cm, AB = 25 cm, and BC = 26. The kite is not drawn to scale.

408 cm2

Suppose C = 44°, a = 6 and b = 9. Find A if a is opposite A.

41.7°

Campsites E and F are on opposite sides of a lake. A survey crew made the measurements shown on the diagram. What is the distance between the campsites? The diagram is not to scale.

41.8

Find the value of x. 19 88 130 42

42

A sphere has a surface area of 900π ft2. Find the volume of the sphere.

4500π ft3

A machinist drilled a cone-shaped hole into a solid cube of metal as shown. If the cube's sides have a length of 4 centimeters, what is the volume of the metal cube after the cone is drilled? Use 3.14 for pi and round your answer to the nearest tenth.

47.3 cm3

Find the volume of the cylinder in terms of pi. 24π in.3 48π in.3 56π in.3 288π in.3

48π in.3

If the ratio of the radii of two spheres is 7 : 2, what is the ratio of the surface areas of the two spheres?

49 : 4

A design on the surface of a balloon is 9 cm wide when the balloon holds 62 cm3 of air. How much air does the balloon hold when the design is 18 cm wide?

496 cm3

A.4.4 B.18.3 C.5.7 D.1.4

5.7

Find the area of the triangle with ∠A = 86, b = 4 ft, and c = 3 ft. Round your answer to two decimal places.

5.99 ft2

Find the slant height of the right cone.

50 m

Find the area of a regular heptagon with side 4 cm.

58.1 cm2

30.2 degrees 35.3 degrees 59.8 degrees 54.7 degrees

59.8 degrees

Describe in words the translation represented by the vector <6, -3>.

6 units to the right, 3 units down

Find the value of x. 6.7 5.4 12.4 11.6

6.7

Eddie started running from his home, and headed west for 4 miles. Then he turned and headed northwest at an angle of 70° with respect to west. If he stayed on this road for 7 miles, how far away from home was he if he could take a straight path to his home?

6.8

Find the area of a regular pentagon with side 6 cm.

61.9 cm2

The area of a regular octagon is 25 cm2. What is the area of a regular octagon with sides five times as large?

625

Solve the problem below. 32 64 45 53

64

25 degrees 45.1 degrees 65 degrees 72.6 degrees

65 degrees

Solve the problem below. 51° 65° 33° 115°

65°

The areas of two similar triangles are 49 cm2 and 16 cm2. What is the ratio of the corresponding side lengths? Of the perimeters?

7 : 4; 7 : 4

To find the height of a pole, a surveyor moves 150 feet away from the base of the pole and then, with a transit 3 feet tall, measures the angle of elevation to the top of the pole to be 26°. What is the height of the pole? Round your answer to the nearest foot.

76

Find the value of x. 79 39 99 159

79

The volumes of two similar solids are 2197 m3 and 64 m3. The surface area of the larger one is 845 m2. What is the surface area of the smaller one?

80 m2

Find the volume of the prism. 40.5 m3 162 m3 9 m3 81 m3

81 m3

Dorothy ran 6 times around a circular track that has a diameter of 47 m. Approximately how far did she run? Use π = 3.14 and round your answer to the nearest meter.

885 m

89.6π m2 57.6π m2 73.6π m2 60.8π m2

89.6π m2

Robert lives in Middleburg. He wants to find the distance between Easton and Westerville. He knows that Easton is 5.7 miles from Middleburg, that Westerville is 9.3 miles from Middleburg, and that the angle formed by the two lines (from Middleburg to Westerville and from Middleburg to Easton) is 69.4°. Rounded to the nearest tenth of a mile, what should Robert find as the distance between Easton and Westerville?

9

If the ratio of the radii of two spheres is 3 : 10, what is the ratio of the surface areas of the two spheres?

9 : 100

Neil had a job helping a jeweler. He had the assignment of counting the faces, vertices, and edges on the emeralds. On the first emerald, Neil counted 9 faces and 16 edges. He quickly realized he didn't have to count the vertices. How many vertices were there?

9 vertices

Find the diameter of the circle. BC = 15 and DC = 19. Round your answer to the nearest tenth. (The diagram is not drawn to scale.) 9.1 6.4 7.2 39.1

9.1

The width of a golden rectangle is 6 m, which is shorter than the length. What is the length?

9.71 m

Find the ratio of the perimeter of the larger rectangle to the perimeter of the smaller rectangle.

9/7

right

90 degrees

Solve the problem below. 80° 120° 90° 100°

90°

Suppose a triangle has three sides which measure 10, 13, and 17. What is the measure of the largest angle?

94.4°

A gardener is making a triangular garden. One angle of the garden is 38° and the sides that surround it are 39.5 feet and 78.5 feet. What is the area of the garden plot?

954.5 ft2

Describe the translation 7 units to the left, 12 units up using a vector.

<-7, 12>

definition of complementary angles

<A and <B are complemtary if m<A+m<B=90

definition of supplementary angles

<A and <B are supplementary if m<A+m<B=180

definition of a right angle

<A is a right angle if m<A=90

What is the formula for trapezoid

A + (b1 + b2) * h) 2

What is the formula for triangle?

A = 1/2 * B * h

What is the formula for square?

A = S (squared)

What is the formula for area of a circle?

A = TTr (squared)

What is the formula for parallelogram?

A = b * h

What is the formula for rectangle

A = b * h

Axiom

A basic assumption that is accepted without proof.

line

A basic undefined term of geometry. A line is made up of points and has no thickness of width. In a figure, a line is shown with an arrowhead at each end. Lines are usually named by lowercase script letters or by writing capital letters for two points on the line, with a double arrow over the pair of letters.

Polygon

A closed plane figure formed by three or more segments that meet endpoint to endpoint.

If-Then Form

A conditional statement written in if-then form uses words "if" and "then".

Point (description)

A dot used in Geometry to indicate a location that has no size, the dimensions are considered 0, and points make up all geometric figures.

foundation drawing

A drawing that shows the base of a structure and tells the height of each part.

Angle

A figure formed by two rays with a common endpoint

Angle

A figure made up of two sides, or rays, with a common endpoint.

What is a plane?

A flat surface that has no thickness

Plane

A flat surface that has no thickness and extends forever

Minimums postulate

A line contains at least two points; a plane contains at least three points all in one line; space contains at least four points all in one plane.

Line

A line is named by identifying any two points on it of by a single lower case letter. Infinite set of points arranged end-to-end in two opposite directions.

Line segment

A line segment consists of two endpoints, and all other points between them.

Transversal

A line that intersects two coplanar lines in two different points; Draw a picture:

Segment bisector

A line that passes through the midpoint of a segment

A perpendicular bisector

A line that passes through the midpoint of the segment and is perpendicular to that segment. Any point on a perpendicular is equidistant from the endpoints of the segment

segment bisector

A line, ray, or segment that cuts another segment in half at the midpoint

perpendicular bisector

A line, ray, or segment that cuts another segment in half at the midpoint and makes 90 degree angles.

Point

A location in space represented by a dot.

What is a point?

A location in space that has no size.

Corresponding Parts

A pair of the sides/angles that have the same relative position in two congruent or similar figures.

ray

A part of a line, with one endpoint, that continues without end in one direction.

Equidistant

A point A is equidistant from B and C if and only if the distance between A and B is the same as the distance between A and C

What is a midpoint?

A point that divides a line segment into two congruent segments.

convex

A polygon in which all vertices appear to be pushed outward.

concave

A polygon in which at least one vertex appear to be pushed inward.

Regular Polygon

A polygon that is BOTH equilateral and equiangular.

Equiangular Polygon

A polygon whose interior angles are all congruent.

Equilateral Polygon

A polygon whose sides are all congruent.

Octagon

A polygon with eight sides.

Undecagon

A polygon with eleven sides.

Pentagon

A polygon with five sides.

Quadrilateral

A polygon with four sides.

Nonagon

A polygon with nine sides.

Heptagon

A polygon with seven sides.

Hexagon

A polygon with six sides.

Decagon

A polygon with ten sides.

Triangle

A polygon with three sides.

Dodecagon

A polygon with twelve sides.

angle bisector

A ray that divides an angle into two congruent angles.

Median of a Triangle

A segment from one vertex of the triangle to the midpoint of the opposite side

Midpoint of a segment

A segment is cut in half by this and will make two segments congruent to each other.

Midsegment of a Triangle

A segment that connects the midpoints of the two sides of a triangle

Radius (of a regular polygon)

A segment with one endpoint at a vertex, the other endpoint at the center of the polygon.

Apothem (of a regular polygon)

A segment with one endpoint on the center of the polygon and that is perpendicular to the midpoint of a side.

Perpendicular Bisector

A segment, ray, line or plane that is perpendicular to a segment at its midpoint

Proof

A sequence of statements and justifications that follow logically from one another. The proof ends when one arrives at the conclusion of the given premises

What is a line?

A series of points that goes on in both directions without end

Line

A set of many points that extend in opposite directions without ending.

Ray

A set of points that is part of a line with one endpoint and extends in one direction with no end.

Postulate

A statement about geometric figures accepted as true without proof.

Postulate

A statement that is accepted as true

Line

A straight path that has no thickness and extends forever

Ray

A subset of a line. Is named by its endpoints listed first and one other point.

Theorem

A theorem is a statement that can be proven by logical deduction

Compass

A tool used to draw circles and parts of circles called arcs.

Protractor

A tool used to draw or measure angles.

Isosceles Triangle

A triangle with at least 2 congruent sides

Isosceles Triangle

A triangle with at least two congruent sides

Scalene Triangle

A triangle with no congruent sides

Equiangular

A triangle with three congruent angles

Equilateral Triangle

A triangle with three congruent sides

Isosceles Triangle

A triangle with two sides of equal length.

Plane

A two-dimensional flat surface.

Conditional

A type of logical statement that has two parts, a hypothesis and a conclusion.

Coordinate Proof

A type of proof that involves placing geometric figures in a coordinate plane

Degree

A unit of angle measure.

Perpendicular lines

A vertical line and a horizontal line that intersect to form 90 degree angles.

Distance between two points on a number line.

AB= |A-B|

x and y are values on a number line.

AB= |x+y|

m<A<90 degrees

Acute

Measures less than 90 degrees

Acute angle

Equiangular

All angles are equal

Acute triangle

All angles less then 90

Equilateral

All three sides equal

Interior Angle of a Polygon

An angel that is formed inside a polygon by two sides meeting at a vertex.

Angle bisector

An angle can be cut in half by a ray called ___________. The two resulting angles are congruent.

Exterior Angle of a Polygon

An angle formed by a side and an extension of an adjacent side of a polygon.

obtuse angle

An angle that is more than 90 degrees

What is a right angle?

An angle that measures exactly 90 degrees.

What is an obtuse angle?

An angle that measures greater than 90 degrees.

What is an acute angle?

An angle that measures less than 90 degrees.

Straight angle

An angle whose measure is 180 degrees.

Right Angle

An angle whose measure is 90 degrees.

Acute angle

An angle whose measure is greater than 0 degrees and less than 90 degrees.

Obtuse angle

An angle whose measure is greater than 90 degrees but less than 180 degrees.

Central Angle of a Polygon

An angle whose vertex is on the center of the polygon and whose rays each pass through consecutive vertices of the polygon.

Postulate

An axiom of geometry

Plane

An infinite set of points arranged as a flat surface. Named by identifying at leas three scattered points. A roman numeral or a single capital letter.

point

An undefined term of geometry. A point is a location. In figures, points are represented by a dot. Points are named by capital letters.

Angle Addition postulate

Angle Addition Postulate states that if a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR.

Vertex Angle

Angle formed by the legs of an isosceles triangle

Vertical angles

Angles that have a common vertex and whose sides are formed by the same lines.

Adjacent angles

Angles with a common vertex and one common side.

Reflexive Property of Congruence

Any figure is congruent to itself, for any segment AB, AB is congruent to AB

Supplementary

Any two angles that total 180 degrees.

Complementary angles

Any two angles that total 90 degrees

Opposite Rays

Are two collinear rays with the same endpoints. Always forms a line.

Amount that fits inside a 2-D shape

Area

Find the measure of B and C. Round the measure of angles to the nearest tenth.

B = 58.1°, C = 81.9°

Between

B is between A and C if and only if all 3 points lie on the same line

Obtuse angle

Between 90 and 180 degrees

Name a point with a ________.

Capital letter

the points of intersection of the medians of a triangle

Centroid

Distance around a circle

Circumference

Two or more lines are concurrent if they intersect at a single point

Concurrent

Alternate exterior are

Congruent

Alternate interior angles are

Congruent

Alternate interior are

Congruent

Corresponding Angles are

Congruent

Line (description)

Consists of points and extends in opposing direction forever

CPCTC Theorem

Corresponding parts of congruent triangles are congruent.

Which figure is a net for a cube?

D

Describe the cross section.

D. pentagon

Distance across a circle

Diameter

Protractor postulate

Each angle has exactly one measure from 0 to 180 degrees

Legs of a Triangle

Either of the sides in a right triangle opposite to the acute angles leg

two vectors are equal if they have the same magnitude and direction; Idea allows us to "slide" any vector

Equal Vectors

What does congruent mean?

Equal in size or length.

All 3 sides are equal

Equilateral triangle

circumcenter, orthocenter, and centroid The endpoints of the segment is the orthocenter and circumcenter

Euler Segment

If the ratio of the radii of two spheres is 2:3, the ratio of their surface areas will be 8:27.

False

The locus "all points equidistant from points A and B" would be a single point directly between points A and B.

False

The reflection of a graph across an axis will always touch its mirror in at least one point.

False

The surface area is found by multiplying the radius of the sphere by 4π.

False

The volume of a prism is the sum of the area of the base and the area of the sides.

False

The volume of this pyramid is 3150 cubic feet.

False

Reflexive Property of Equality

For any real number a, a=a

law of detachment

Given P → q and p true. Conclude: q

law of syllogism

Given p→q and q→r Conclude: p→r

Congruent

Having the same size and shape

Transitive Property

IF a=b and b=c, then a=c

Symmetric Property

IF a=b, then b=a

angle addition postulate

Id D is in the interior of <ABC, then m<ABD+m<DBC=m<ABC

Law of Detachment

Identify the hypothesis of the conditional, then you can make a valid conclusion. pg.113

Third Angle Theorem

If 2 angles of 1 triangle are congruent to 2 angles of a second triangle then the third angles of the triangles are congruent

Segment addition postulate

If A-B-C, then AB+BC=AC

Segment Addition Postulate

If B is between A and C, then AB+BC=AC

Transitive Property of Equality

If a = b and b = c, then a = c

division property of equality

If a = b and c ≠ 0, then a/c=b/c

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle

Substitution Property of Equality

If a=b, then a can be substituted for b in any equation or expression

Addition Property of Equality

If a=b, then a+c=b+c

Subtraction Property of Equality

If a=b, then a-c=b-c

Multiplication Property of Equality

If a=b, then ac=bc

SAS Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

Concurrency of Angles Bisectors of a Triangle

If angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

Inverse

If not p then not q. ~p------> ~q

Contropositive

If not q then not p. ~q------> ~p

Bi-Conditional

If p then q and if q then p or p <-------> q. p if and only if (iff.) q

Law of Syllogism

If p then q and q then r are trye, then p then r is also true.

Law of Detachment

If p then q is a true biconditional statement & p is true then q is true.

Conditional Statement

If p then q. It is a type of logical statement that has two parts, a hypothesis (p) and a conclusion (q). p -------> q

angle addition postulate

If point B lies in the interior of angle AOC then the measure of angle AOB + the measure of angle BOC = the measure of angle AOC. If AOC is a straight angle and point B is any point not on line AC, then the measure of angle AOB + the measure of angle BOC = 180.

Converse

If q then p. q --------> p

HL

If the hypotenuse and a let of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

Half planes

If the rays of an angle are opposite rays, then the opposite rays form a line that divides the plane into two half planes

SSS

If three sides of one triangle are congruent to three sides of a second triangle then the two triangles are congruent

AAS

If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

ASA

If two angles and the included side of one triangle are congruent to two angles and the included sides of a second triangle, then the two triangles are congruent.

ASA Theorem

If two angles and the included side of one triangle are equal to two angles and the included side of another triangle then the triangles are congruent

Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are also congruent

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Theorem 1-1

If two lines intersect, the they intersect in exactly one point.

Theorem 1-3

If two lines intersect, then exactly one plane contains the lines

Postulate 9

If two planes intersect, the their intersection is a line

Postulate 8

If two points are in a plane, then the line that contains the points is in that plane

SAS

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, the the two triangles are congruent.

Base Angles Theorem

If two sides in a triangle are congruent, then the angles opposite to the sides are also congruent.

Perpendicular Bisector Theorem

In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Hypotenuse

In a right triangle, the side opposite the right angle.

Isosceles Triangle Theorem

In an isosceles triangle, the base angles are congruent to each other.

segment addition postulate

In geometry, the segment addition postulate states that if B is between A and C, then AB + BC = AC

Segment

Is a subset of a line. Has two endpoints and includes all of the other points of a line that fall between those two endpoints.

2 sides are equal

Isosceles triangle

Inverse

It is formed by negating the hypothesis and conclusion of the conditional statement.

Contrapositive

It is formed by switching and negation the hypothesis and conclusion of the conditional statement.

Converse

It is formed by switching the hypothesis and conclusion of the conditional statement.

Negation

It is formed by writing the negative of the statement.

Solve the problem below. JK = 3, KL = 7 JK = 10, KL = 9 JK = 6, KL = 6 JK = 9, KL = 10

JK = 9, KL = 10

Q.E.D.

Latin: "quod erat demonstrandum" "That which was to be demonstrated"

Acute angle

Less than 90 degrees

Perpendicular lines

Lines that form right angles.

is the length of the arrow from A to B in a vector and is denoted by AB you can use the Pythagorean theorem and the distance formula to find the length

Magnitude

What do we use a protractor for?

Measuring angles.

Scalene triangle

N0 sides equal

m<A>90

Obtuse

Measures between 90 degrees and 180 degrees

Obtuse angle

Obtuse triangle

One angle more then 90 degrees

What are vertical angles?

Opposite angles formed by two lines- think of the X !

Vertical angles

Opposite angles. They form an "x" with their rays and are always congruent.

Arc

Part of a circle.

What is a line segment?

Part of a line (a segment has two endpoints).

Line segment

Part of a line between two points.

What is a ray?

Part of line that has one endpoint and goes on forever in the other direction.

Distance around a polygon

Perimeter

Three undefined terms

Point Line Plane

Betweenness

Point B is between points A and C if: 1) Points A, B, and C are collinear 2) The distance from A to C is = to the sum of the distances A to B and B to C "A-B-C"

What are collinear points

Points on the same line

Collinear

Points on the same line.

What does coplanar mean?

Points or lines that are on the same plane.

Collinear points

Points that all lie on the same line.

Coplanar points

Points that lie in the same plane and are capable of being contained by the same plane.

collinear points

Points that lie on the same line

collinear points

Points that lie on the same line.

A 2-D shape made up of 3 or more straight lines

Polygon

vectors starting at the origin

Position vectors

Distance halfway across a circle

Radius

Ray AB

Ray AB consists of segment AB and all points C such that B is between A and C. (Point A is called the endpoint of the Ray).

Angle bisector

Ray that divides an angle into two equal parts.

Included Angle

Relationship involving two sides and the angle they form

Measures 90 degrees

Right angle

All sides are different lengths

Scalene triangle

Corresponding Parts

Sides or angles that occupy the same relative positions in similar polygons.

What is a postulate?

Something so obvious that we don't need to prove it. (Angle Addition Postulate and the Segment Addition Postulates are the two that we know)

Measures 180 degrees

Straight angle

Triangle Sum Theorem

Sum of the measures of the interior angles of a triangle is 180 degrees

Same side interior are

Supplementary or equal to 180

Acute Angles of a Right Triangle Corollary

The acute angles of a right triangle are complementary

Adjacent angles

The angles "next to" each other. The share a common side and a vertex.

Vertical angles

The angles opposite each other when 2 lines intersect. They share only a vertex

Bisector

The bisector of a segment is a line, segment, ray, or plane that intersects a segment at its midpoint.

Circumference

The distance around a circle

AB

The distance from A to B

n-gon

The generic name given to any polygon if it has more than 12 sides. (Example: A 13-sided polygon is called a 13-gon).

Vertex

The intersection point, marks a change of direction

AB with arrow both ways

The line AB

AB with line over it

The line segment AB

Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent

Measure of a Central Angle (in regular polygons)

The measure of a central angle in a polygon with n number of sides can be found using the formula 360/n. (Example: A pentagon has 5 sides (n = 5), so a central angle in a regular pentagon has a degree measure of 72 (360/5)).

Exterior Angles Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side

Midpoint

The midpoint of a segment is the point that divides the segment into two congruent segments

Base of a Triangle

The non congruent side of an isosceles triangle that only has two congruent sides

Base

The number being multiplied by itself.

Exponent

The number of times a base is multiplied by itself.

Concurrency of Perpendicular Bisectors of a Triangle

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle

Altitude of a Triangle

The perpendicular segment from one vertex of the triangle to the opposite side or to the line that contains the opposite side

Coordinate Plane

The plane formed by 2 number lines that intersect at their zero points dividing the plane into 4 quadrants

Orthocenter

The point at which the lines containing the three altitudes of the triangle intersect

intersection

The point at which two or more lines or cross. The line in which two or more planes have in common.

Vertex

The point common to both sides of an angle.

Incenter

The point of concurrency of the three angle bisectors of the triangle

Circumcenter

The point of concurrency of the three perpendicular bisectors of the triangle

Centroid

The point of concurrency pod the three medians of the triangle

Point of Concurrency

The point of intersection of concurrent lines, rays or segments.

Midpoint of a Segment

The point that divides the segment into two congruent segments

Ruler postulate

The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers

Definition

The precise meaning of a term

Compare the quantity in Column A with the quantity in Column B. The quantity in Column A is greater. The quantity in Column B is greater. The two quantities are equal. The relationship cannot be determined on the basis of the information given.

The quantity in Column A is greater.

AB with one arrow over

The ray AB

Equidistant

The same distance from one figure as from another figure

Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

Height

The segment that is perpendicular to the base

converse

The statement formed by exchanging the hypothesis and conclusion of a conditional statement. Pg. 101

Geometry

The study of points, lines, surfaces, and solids

180°

The sum of adjacent angles

Polygon Angle Sum Formula

The sum of all the interior angles of a polygon with n number of sides can be found with the formula: 180(n - 2).

Polygon Exterior Angle Sum Theorem

The sum of the measures of one set of exterior angles of any polygon is 360 degrees.

Base Angles of an Isosceles Triangle

The two angles in an isosceles triangle that are opposite of the congruent sides.

How do you name a plane?

The word "Plane"... then three non-collinear points.

Opposite rays

They share the same endpoint, and point in opposite directions.

isometric drawing

This is a 2 dimensional drawing of a 3 dimensional object that shows all dimensions.

orthographic drawing

This is a 2 dimensional drawing of a 3 dimensional object using 3 pictures, each showing one pair of dimensions

conjecture

This is a conclusion you reach using inductive reasoning.

counterexample

This is an example for which a conjecture is incorrect.

inductive reasoning

This is based on patterns that you observe.

Concurrent

Three or more lines, rays or set,emts that intersect in the same point

theorem 1-2

Through a line and a point not on the line there is exactly one plane.

Postulate 7

Through any three there is at least one plane, and the through any three noncollinear points there is exactly one plane

Postulate 6

Through any two points there is exactly one line

Bisect

To divide into two congruent parts

Bisect

To divide into two equal parts.

Intersect

To meet at a point; to cross or overlap each other.

Right triangle

Triangle with a right angle

A line that intersects the radius of a circle at a point on the circle and is perpendicular to the radius is known as a tangent.

True

All circles have rotational symmetry.

True

The measure of a major arc will always be greater than that of a semicircle.

True

The measure of angle CAB is 44 degrees.

True

The measure of angle COB is 82 degrees.

True

Adjacent angles

Two angles are adjacent if and only if: 1) They share a common vertex 2) They share a common side 3) They do not have common interior points 4)and if they are coplanar

Definition of Adjacent Angles

Two angles are adjacent if they share a endpoint and a common ray

Vertical angles

Two angles such that the sides of one angle is opposite to the other angle. They are congruent

Linear pair

Two angles the form a straight line totaling 180

Supplementary angles

Two angles whose measures add to 180 degrees.

Complementary angles

Two angles whose measures add to 90 degrees.

Complementary Angles

Two angles whose sum is 90 degrees

Coplanar points

Two are more points are coplanar iff they are on the same plane

Congruent Figures

Two geometric figures that have exactly the same size and shape. When two figures are congruent, all pairs of corresponding angles and sides are congruent.

Congruent

Two geometrical figures that have the same shape and same size are congruent.

Parallel lines

Two lines that will never intersect

What are supplementary angles?

Two or more angles that add up to exactly 180 degrees. Windshield wiper angles!

What are complementary angles?

Two or more angles that add up to exactly 90 degrees.

Collinear points

Two points are collinear iff they are on the same line

What is an angle?

Two rays with the same endpoint.

Isosceles triangle

Two sides equal

plane

Undefined term of geometry. A fat surface that extends indefinitely in all directions. Represented by a shaded, slanted 4 sided figure. Named by a capital script letter or by three noncollinear points on the plane

How do you name a line segment?

Use the two endpoints and the segment symbol: ___

Flow Proof

Uses arrows to show the flow of the logical argument

contains quantities that can be described both by their magnitude and direction. Such a quantity is called a vector

Vector

Vertical angles theorem

Vertical angles are congruent

Reflectional Symmetry

When a reflection occurs and the image appears to be unchanged. Some objects can be reflected using more than one line of reflection and still be symmetrical.

Rotational Symmetry

When a rotation occurs and the image appears to be unchanged. A figure must have a minimum of 2 rotations to have rotational symmetry.

Symmetry

When the pre-image undergoes a transformation and the new image looks exactly like the pre-image. It appears as though no change has taken place. This can occur using a reflection or a rotation.

How do you name a ray?

Write the endpoint first and then another point on the ray. Use the ray symbol: ->

Which letter has rotational symmetry?

X

reflexive property of equality/congruence

XY equals XY or XY is congruent to XY

Are the polygons similar? If they are, identify the correct similarity statement and ratio. Yes, ΔABC is similar to ΔDEF with a similarity ratio of 2 : 3. Yes, ΔABC is similar to ΔFED with a similarity ratio of 21 : 11. Yes, ΔABC is similar to ΔDEF with a similarity ratio of 3 : 2. The figures are not similar.

Yes, ΔABC is similar to ΔDEF with a similarity ratio of 3 : 2.

Which description would NOT guarantee that the figure was a square? A. A parallelogram with perpendicular diagonals B.Both a rectangle and a rhombus C.A quadrilateral with all sides and all angles congruent D.A quadrilateral with all right angles and all sides congruent

a

A circle has radius 6 cm. In the plane of the circle, what best describes the locus of points that are 2 cm from the circle?

a circle of radius 4 cm and a circle of radius 8 cm

polygon

a closed plane figure made up of line segments

theorem

a conjecture that has been proven

angle

a figure formed by two rays with a common endpoint

plane

a flat surface that goes on forever in all directions

Point

a geometric element that has position but no size or shape

angle

a geometric object formed by two rays with a common endpoint

conjecture

a hypothesis or theory. an educated guess made by looking at information

Intersection of two planes

a line

transversal

a line that intersects 2 (or more) other lines at distinct points; all the lines lie in the same plane.

transversal

a line that intersects two lines at distinct points is a __________

transversal

a line that intersects two or more coplanar lines at different points

perpendicular bisector

a line that is perpendicular to a segment at its midpoint

point

a location. It has no size.

Degree

a measure for arcs and angles

vertical angles

a pair of opposite congruent angles formed by intersecting lines

rectangle

a parallelogram with 4 right angles

rhombus

a parallelogram with all sides the same length

ray

a part of a line that starts at one endpoint and extends forever

Intersection of two lines

a point

endpoint

a point at one end of a segment or the starting point of a ray

Endpoint

a point at one end of a segment or the starting point of a ray.

Midpoint

a point in the middle of a segment

definition of midpoint

a point is a midpoint if it cuts a segment into 2 congruent segments

Midpoint

a point that divides a segment into two congruent segments

midpoint

a point that divides a segment into two congruent segments

bisector

a point, line, plane or ray that divides a figure in half

regular polygon

a polygon that is both equilateral and equiangular

triangle

a polygon with 3 sides

triangle

a polygon with 3 sides and 3 angles

quadrilateral

a polygon with 4 sides and 4 angles

n-gon

a polygon with n sides

pyramid

a polyhedron that has any polygonal base and one vertex opposite that base, all the others faces are triangles

prism

a polyhedron with two parallel, congruent faces called bases, all the other faces are rectangles

rhombus

a quadrilateral with four congruent sides

rectangle

a quadrilateral with four right angles

square

a quadrilateral with four right angles and four congruent sides

trapezoid

a quadrilateral with one pair of parallel sides

parallelogram

a quadrilateral with opposite sides the same length and parallel to each other

kite

a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent

parallelogram

a quadrilateral with two pairs of parallel sides

definition of angle bisector

a ray is an angle bisector if it cuts an angle into 2 congruent angles

Angle bisector

a ray that cuts an angle into 2 equal (congruent) parts

Angle bisector

a ray that divides an angle into two congruent angles is called and angle bisector

side

a ray that forms an angle

Angle Bisector

a ray that splits an angle into 2 congruent adjcent angles.

square

a rectangle with all sides the same length

The graph of a pentagon is in Quadrant I. Describe a reflection that will result in a pentagon in Quadrant III.

a reflection in the line y = -x

midsegment of a trapezoid

a segment that connects the midpoint of the two non parallel sides of a trapezoid

Pythagorean triple

a set of three whole numbers that satisfy the Pythagorean Theorem

polygon

a simple, closed figure formed by three or more line segments

Corollary

a statement that can be easily proved using a theorem

corollary to a theorem

a statement that can be proved easily using the theorem

theorem

a statement that can be proven using defined terms and statements we know are true.

counterexample

a statement that proves a conditional statement wrong

line

a straight path that extends without end in opposite directions

sphere

a three dimensional figure formed by all the points that are the same distance from the center of the sphere

polyhedron

a three dimensional figure formed by polygons

cone

a three dimensional figure that has one circular base and a vertex

cylinder

a three dimensional figure that has two parallel, congruent bases that are circles

When Mario has to leave the house for a while, he tethers his mischievous puppy to the corner of a 12 ft-by-8 ft shed in the middle of his large backyard. The tether is 18 feet long. Which description fits the boundary of the locus of points in the yard that the puppy can reach?

a three-quarter circle of radius 18 ft, quarter circles of radii 10 ft and 6 ft

Triangle

a three-sided polygon

isosceles trapezoid

a trapezoid whose nonparallel sides are congruent

isosceles triangle

a triangle with 2 or more sides the same length

acute triangle

a triangle with 3 acute angles

Acute Triangle

a triangle with 3 acute angles (less than 90 degrees)

scalene triangle

a triangle with 3 sides of different lengths

Equiangular Triangle

a triangle with all angles congruent

equilateral triangle

a triangle with all sides the same length

isosceles triangle

a triangle with at least 2 congruent sides

Isosceles

a triangle with atleast 2 congruent sides

Scalene

a triangle with no congruent sides

Obtuse Triangle

a triangle with one obtuse angle

obtuse triangle

a triangle with one obtuse angle

Right Triangle

a triangle with one right angle

right triangle

a triangle with one right angle

acute triangle

a triangle with three acute angles

Equilateral Triangle

a triangle with three congruent sides

Equilateral

a trienagle with all congruent sides

Deductive Reasoning

a type of reasoning in which the conclusion is based on accepted statements.

Inductive Reasoning

a type of reasoningin which the conculsion is based on several past observations

Distributive Property of Equality

a(b+c)=ab+ac

distributive property of multiplication

a(b+c)=ab+ac ab+cb=(a+c)b *used to combine like terms

CD = 60, OM = 18, and ON = 15. Find FN.

a. 3 111

Refexive Property

a=a pg.121

posulate/axiom

accepted statement of fact

Are two angles that:Have a common vertex and common side BUT have no common interior points

adjacent angles

equiangular

all angles are congruent

right angle congruence theorem

all right angles are congruent

equilateral

all sides are congruent

Law of Syllogism

allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement. pg.115

An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side

altitudes

postulate

an accepted statement of fact

exterior angle

an angle formed by one side of a polygon and the extension of an adjacent side

interior angle

an angle inside a polygon

right angle

an angle that is exactly 90 degrees

acute angle

an angle that is less than 90 degrees

right angle

an angle that measures 90 degrees

acute angle

an angle that measures between 0 and 90 degrees

obtuse angle

an angle that measures between 90 and 180 degrees

straight angle

an angle that measures exactly 180 degrees (a straight line)

straight angle

an angle that measures exactly 180°

right angle

an angle that measures exactly 90 degrees

right angle

an angle that measures exactly 90°

reflex angle

an angle that measures greater than 180° but less than 360°

acute angle

an angle that measures less than 90 degrees

acute angle

an angle that measures less than 90°

obtuse angle

an angle that measures more than 90°

straight angle

an angle whose measure is exactly 180 degrees, a straight line

acute angle

an angle whose measure is less than 90 degrees

obtuse angle

an angle whose measure is more than 90 degrees but less than 180 degrees

Corollary

an equilateral triangle has angles that measure 60.

point

an exact location represented by a dot

counterexample

an example for which the conjecture is incorrect

counterexample

an example that proves a statement false

Counterexample

an example that shows a conjecture is not always true

conditional statement

an if-then statement that is used in definitions (i.e. If I go to math class then I will learn math)

conjecture

an unproven statement that is based on observations (like a hypothesis)

Part A: if B lies in the interior of <AOC then : M< AOB + M<BOC =M<AOC

angle addition postulate

A ray that consist of a set of all points equidistant from the two sides of the angle

angle bisector

A ray that consist of a set off all points equidistant from the two sides of a triangle

angle bisectors

AIA (alternate interior angles)

angles are on opposite sides of the transversal and on the inside of the given lines

corresponding angles

angles formed by a transversal cutting through 2 or more lines that are in the same relative position

base

angles on an isosceles triangle that include the base and are cngruent are called __________ angles

Supplementary angles

angles that add up to 180 degrees

Complementary angles

angles that add up to 90 degrees

CIA (consecutive interior angles)

angles that are on the same side of the transversal and inside the two lines

AEA (alternate exterior angles)

angles that lie outside the two lines and on opposite sides of the transversal

congruent angles

angles with the same measure

adjacent angles

are a pair of angles with a common vertex and a common side, but no common interior points

Which statement is NOT true? a = 51° b = 103° d = 39° c = 78°

b = 103°

b = 14.89, C = 77°, c = 17.11 b = 12.53, C = 77°, c = 4.83 b = 12.53, C = 77°, c = 13.3 b = 14.89, C = 77°, c = 16.8

b = 12.53, C = 77°, c = 13.3

Obtuse angle

between 180 and 90 degrees

obtuse

between 90 and 180 degrees

Which describes the locus of points 4 units from (5, 1) in the coordinate plane?

circle with center (5, 1), diameter 8

the point of intersection of the perpendicular bisectors

circumcenter

Polygon

closed plane figure with straight sides without intersecting sides

The ratio of the segments into which the altitude to the hypotenuse of a right triangle divides the hypotenuse is 9 : 4. What is the length of the altitude?

cnbd

vertex

common endpoint of an angle

If a quadrilateral is a parallelogram, then its opposite sides are

congruent

Angles that have equal measures

congruent angles

vertical angles

congruent angles opposite one another, formed by intersecting lines

Segments that are the same size and shape

congruent segments

parallel lines

coplanar lines that do not intersect

Segment bisector

cuts a segment into 2 congruent parts

bisector

cuts something into two equal parts

circumference

distance around the outside of a circle

If you know the diameter of a circle, how do you find the radius?

divide by 2

Vertical Angle

either of two equal and opposite angles formed by the intersection of two straight lines

base angles of trapezoid

either pair of angles whose common side is a base of a trapezoid

congruent

equal in shape and size

Straight angle

exactly 180 degrees

Right angle

exactly 90 degrees

A vector sum cannot show the result of vectors that act at the same time.

false

An isometry is a transformation in which the preimage and image are not congruent.

false

An isosceles right triangle's hypotenuse is twice as long as its leg.

false

Any angle bisector will divide the opposite side into two equal parts.

false

For two figures to be similar, they must be arranged in the same orientation.

false

Given the law of sines, all the measures of a triangle can be found with two sides and any angle.

false

Some rectangles do not have a line of symmetry.

false

The Law of Cosines states that a2 = b2 - c2 + 2bc cos A.

false

The angle formed by the intersection of a secant and a tangent is equal to the sum of the measures of the intercepted arcs.

false

The area of a circle with a radius of 7 cm is approximately 158.36 square centimeters.

false

The area of a regular hexagon is 14 cm2. The area of a regular hexagon with sides twice as large is 28 cm2.

false

The hypotenuse of a right triangle will always be adjacent to the right angle.

false

The surface area of a pyramid is the sum of the areas of the triangles making up its sides.

false

Two triangles are similar only if they share a congruent angle and two congruent sides adjacent to the angle.

false

Congruent Figures

figures that have the same size and shape

plane

flat surface that has no thickness.

plane

flat surfce that goes on forever

angle

formed by two rays with the same endpoint

same-side interior angles

found inside the lines on the same side of a transversal. these angles are supplementary

same side exterior angles

found on the outside of the two lines on the same side of a transversal. These angles are supplementary

alternate exterior angles

found outside the two angles on opposite sides these angles are congruent

line

goes on forever in both directions

point

has no dimensions, only location

congruent

having the same measure

congruent complements theorem

if 2 angles are complementary to the same angle, then they are congruent. if 2 angles are complementary to congruent angles, then they are congruent.

complementary/supplemenatry theorem

if 2 angles are completmentary (supplementary) to same (or congruent) angles then they are congruent

congruent supplements theorem

if 2 angles are supplementary to the same angle, then they are congruent. if 2 angles are supplements to congruent angles, then they are congruent.

linear pair theorem

if 2 angles create a linear pair then they are supplementary

linear pair postulate

if 2 angles form a linear pair, then they are supplementary

3rd triangle theorem

if 2 angles in 1 tirangle are congruent to 2 angles in another triangle then the 3rd triangle must also be congruent

converse of a bisecting line theorem

if 2 line from right angles then the lines are bisecting

exterior side of bisecting lines theorem

if 2 sides of adjacent angles form bisecting lines then angles are complementary

Segment addition postulate

if B is between A and C, then AB+BC=AC

corollary to the base angles theorem

if a triangle is equilateral then its equiangular

transitive property of equality/congruence

if a=b and b=c, then a=c (same for congruent signs)

Division Property of Equality

if a=b and c≠0, then a/c=b/c

substitution property of equality

if a=b, then a may be substituted for b in any expression

addition property of equality

if a=b, then a+c=b+c

subtraction property of equality

if a=b, then a-c=b-c

multiplication property of equality

if a=b, then ac=bc

Symmetric Property of Equality

if a=b, then b=a

symmetric property of equality/congruence

if a=b, then b=a. if a is congruent to b, then b is congruent to a.

Transitive Property of Congruence

if figure a is congruent to figure b and figure b is congruent to figure c, then figure a is congruent to figure c

Symmetric Property of Congruence

if figure a is congruent to figure b, then figure b is congruent to figure a

Angle Addition Postulate

if point B lies in the interior of <AOC then m<AOB+m<BOC=m<AOC

hypotenuse leg congruence theorem

if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent

HL Theorem

if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent

SSS Theorem

if the lengths of the corresponding sides of two triangles are proportional then the triangles are similar

side side side congruence postulate

if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent

AAS Theorem

if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent

angle angle side congruence theorem

if two angles and and a non included side of a triangle are congruent to two angles and the corresponding non included side then the two triangles are congruent

angle side angle congruence postulate

if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle then the two triangles are congruent

Isosceles Triangle Theorem Converse

if two angles of a triangle are congruent the sides opposite them are congruent

converse of the base angles theorem

if two angles of a triangle are congruent then the sides opposite of them are congruent

third angle theorem

if two angles of one triangle are congruent to two angles of another triangle, then the third angle is also congruent

base angles theorem

if two side of a triangle are congruent then the angles opposite them are congruent

side angle side congruence postulate

if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two sides are congruent

Isosceles Triangle Theorem

if two sides of a triangle are congruent, then the angles opposite those sides are congruent

corresponding angles

in a transversal when angles are in the same position on different lines

Base Angles

in an isosceles triangle, the angles adjacent to the base

the point of intersection of the angle bisectors

incenter

line

infinite set of points

perpendicular

intersecting at or forming right angles

conjecture

is a conclusion you reach using inductive reasoning.

theorem

is a conjecture or statement that you prove true. pg 128

proof

is a convincing argument that uses deductive reasoning. pg. 123

biconditional

is a single true statement that combines a true conditional and its true converse. PG. 105

conditional

is an if-then statement. pg. 97

hypothesis

is the part p following if. pg. 97

conclusion

is the part q following then. pg. 97

paragraph proof

is the written as sentences in a paragraph. pg. 130

The distance between two endpoints

length of a segment

Acute angle

less than 90 degrees

acute angle

less than 90 degrees

ray

line segment with 1 endpoint, and only going in one direction.

Line

line that continues without end

Line Segment

line with 2 endpoints

Ray

line with one endpoint and one arrow

parallel

lines in the same plane that do not intersect

intersecting

lines or line segments in the same plane that have one point in common

perpendicular

lines or line segments that intersect at a right angle (90°)

parallel

lines or line segments that lie in the same plane and NEVER intersect. They are always equal distance apart.

perpendicular

lines that intersect to form 90 degree angles

skew lines

lines that lie in different planes and are neither parallel nor intersecting

parallel

lines that never intersect

two column proof

lists each statement on the left. pg. 123

Angle addition postulate

little angle 1 + little angle 2 = total big angle

Segment addition postulate

little segment 1 + little segment 2 = total big segment

coplanar

lying in the same plane

concave polygon

makes a cave. Points go outside shape.

Exterior Angle Theorem

measure of the exterior angle is equal to the sum of the remote interior angles

A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

medians

The point that divides a segment into two congruent parts.

midpoint

inverse

negate both the hypothesis and the conclusion of the conditional. pg 101

contrapositive

negating both the hypothesis and conclusion of the converse statement. Pg. 101

Are the two figures similar? If so, give the similarity ratio.

no

Scalene Triangle

no congruent sides

scalene triangle

no congruent sides

Is line AB tangent to the circle? Why or why not? (The diagram is not to scale.)

no; 9 2 + 12 2 ≠ 14 2

skew lines

noncoplanar; they are not parallel and do not intersect

Find the surface area of the cone in terms of pi. 800π cm2 230π cm2 260π cm2 none of these

none of these

Which regular polygon can be used to form a tessellation? The sum of the measures of the angles of each polygon is given.

none of these

negation

of a statement p is the opposite of the statement. pg.101

the point of intersection of the three altitudes

orthocenter

ray

part of a line consisting of one endpoint and all the points of the line on one side of the endpoint

segment

part of a line consisting of two endpoints and all the points between them

ray

part of a line that has one endpoint and all points in one direction

Ray

part of a line with one endpoint

Segment

part of a line with two endpoints

line segment

part of a line with two endpoints

segment

part of a line with two endpoints

line segment

part of a line with two endpoints. A measurable part of a line. Denoted using two capital letters representing the endpoints with a line(no arrows) over the top of them.

corresponding parts

parts that would match if placed on top of each other `

line ray or segment that is perpendicular to the segment at the midpoint

perpendicular bisectors

parallel planes

planes that do not intersect

midpoint

point that divides a segment into two congruent segments.

coplanar

points and lines in the same plane

coplanar

points lying on the same plane

Collinear

points that lie on the same line

collinear

points that lie on the same line

Coplanar

points that lie on the same plane

coplanar

points that lie on the same plane

A portion of a line with one endpoint

ray

inductive reasoning

reasoning based on patterns you observe. Pg. 90

m<A=90

right

congruent

same shape and same size

Same side Exterior

same side of the transversal, but not corresponding or vertical, just next to eachother

A and B and all points between A and B

segment

If point B is between A and C then :AB+ BC= AC

segment addition postulate

Intersects the segment at its midpoint

segment bisector

median

segment from a vertex to the midpoint of the opposite side in a triangle

Congruent segments

segments with equal lengths

line

series of points that extends in two opposite directions without end.

plane

set of points forming a flat surface, extends infinitely , no thickness

collinear

set of points on the same line

circle

starts with a given point called the center and all the points that are the same distance from the center

equivalent statements

statements that have the same truth value. Pg. 102

CPCTC Postulate

states that if we know that two triangles are congruent, we can conclude that all corresponding angles and all corresponding sides are congruent.

m<A=180

straight

hypothesis

the "if" part of a conditional statement

conclusion

the "then" part of a conditional statement

Corollary

the 2 acute angles in a right triangle must be completmentary

corollary to the triangle sum theorem

the acute angles of a right triangle are complementary

volume

the amount of space that an object holds

vertex angle of isosceles triangle

the angle formed by the congruent sides in an isosceles triangle

vertex angle

the angle formed by the legs

vertex

the angle in an isosceles triangle that is not congruent to the others is the _________ angle

vertex

the common endpoint of two rays that form an angle

legs

the congruent sides of an isosceles triangle

diameter

the distance across a circle through its center

radius

the distance from the center of the circle to any point on the circle

Vertex

the endpoint of an angle

vertex

the endpoint shared by two or more rays in an angle.

Remote interior angles theorem

the exterior angles of a triangle euquals the sum of the 2 remote interior angles.

angle

the intersection of two noncollinear rays at common endpoint. The rays are called sides and the common endpoint is called the vertex. Denoted by using three points on the angle with the vertex point being the middle letter or by a number as in the corresponding picture.

Distance

the length of a path between two points

distance

the length of a path between two points.

distance

the length of the path between two points

exterior angle theorem

the measure of an exterior angle of a triangle is equal to the sum of the measures of two nonadjacent interior angles

protractor postulate simplified

the measure of angle POQ is equal to |x-y| if ray OP is paired with x and ray OQ is paired with y.

Congruent

the measure of vertical (opposite) angles

legs of trapezoid

the nonparallel sides of a trapezoid

area

the number of square units needed to cover a two dimensional figure

converse

the opposite. I.E. flipping the conditional statement around and changing the "if" to the "then" and the "then" to the "if"

altitude

the perpendicular segment from a vertex of a triangle to the opposite side or the line that contains the opposite side

edge

the place where two or more faces meet

midpoint

the point on a segment that divides it into two congruent segments

vertex

the point where two or more edges meet

faces

the polygons that form the solids

deductive

the reasoning used when general statements are combined into specific conclusions

inductive

the reasoning used when you examine a body of information and try to find patterns or relationships

equidistant

the same distance from two or more objects

space

the set of all points

angle

the set of points formed when two rays intersect at their endpoints

intersection

the set of points that two or more geometric figures have in common

hypotenuse

the side opposite the right angle in a right triangle

base of isosceles triangle

the side opposite the vertex angle in an isosceles triangle

base

the side that is not congruent in an isosceles triangle

triangle sum theorem

the sum lf the measures of the interior angles=180 degrees

180°

the sum of all 3 angles in a triangle

360°

the sum of all 4 angles in a quadrilateral

surface area

the sum of areas of all the faces of a three dimensional figure

triangle sum theorem

the sum of the interior andles of a trial is 180

perimeter

the sum of the lengths of the sides

Triangle Sum Theorm

the sum of the measures of the interior angles of a triangle is 180

base

the third side of the isosceles triangle

base angles

the two angles adjacent to the base

base angles of isosceles triangle

the two angles adjacent to the base

legs

the two congruent sides of an isosceles triangle

bases of trapezoid

the two parallel sides of the trapezoid

iff

the word used to notate "if and only if"

solid

three dimensional figure

right triangle

triangle with one right angle

congruent

triangles that are exactly the same are __________ triangles

A circumscribed circle will touch every vertex of a regular polygon.

true

A cross section is the intersection of a plane and a solid figure.

true

A dilation is a transformation whose preimage and image are similar.

true

A proportion is a statement of equality for two ratios.

true

A translation is an isometry that maps all points of a figure the same distance in the same direction.

true

A vector is any quantity with magnitude and direction.

true

If points of a region represent equally-likely outcomes, then you can find probabilities by comparing areas.

true

The Law of Cosines lets you find missing measures in a triangle when you know the measures of two sides and the included angle, or three sides.

true

The area of a pentagon with a perimeter of 50 inches is 172.05 square inches.

true

The center of a circle with the equation (x - 6)2 + (y + 2)2 = 49 is ( 6, -2).

true

The length of a semicircle is half the perimeter of the circle to which it belongs.

true

The surface area of a cylinder with a height of 4 inches and a base radius of 1 inch is approximately 31.4 inches squared.

true

The volume of a cone with a height of 9 inches and a base area of 7 square inches is 21 cubic inches.

true

Two triangles are similar if they share two congruent angles.

true

linear pair

two adjacent angles that form a straight line on one side

Linear Pair

two adjacent angles whose non-common sides are opposite rays

adjacent angles

two angles having the same vertex and sharing one side

complementary

two angles that add to be 90 degrees

linear pair

two angles that are adjacent and supplementary

congruent angles

two angles that have = measure

supplementary angles

two angles who measures have sum of 180 degrees

Supplementary Angles

two angles whose measures have a sum of 180 degrees

supplementary

two angles whose measures have a sum of 180°

complementary angles

two angles whose measures have a sum of 90 degrees

vertical angles

two angles whose sides form two pairs of opposite rays

vertical angles

two angles whose sides form two pairs of opposite rays. pg 38

supplementary angles

two angles whose sum is 180 degrees

complementary angles

two angles whose sum is 90 degrees

opposite rays

two collinear rays with the same endpoint. Opposite rays always form a line.

adjacent angles

two coplanar angles with with a common side, a common vertex, and no common interior points

nets

two dimensional patterns for three dimensional figures

How do you name a line?

two points and the line symbol. <->

opposite ray

two rays going opposite directions that make a line and have the same endpoint.

opposite ray

two rays that have a common endpoint and form a line

congruent segments

two segments with the same length

vertical angle congruence theorem

vertical angles are congruent

bi-conditional statement

when both the conditional statement and its converse are true. this means that instead of "if" and "then" there is "if and only if"

alternate interior angles

when the angles are found on the opposite sides of the interior of the transversal. They are congruent.

Interior Angles

when the sides of a triangle are extended other angles are fromed, these are the 3 original angles of the triangle

Exterior Angles

when the sides of a triangle are extended, these are the angles on the outside of the triangle (or angles that are adjacent to the interior angles)

between

when three points are collinear, you can say that one point is_____the other two

truth value

whether a statement is true or false. pg. 98

The polygons below are similar, but not necessarily drawn to scale. Find the values of x and y.

x = 13, y = 5

Find the value of x if AB = 20, BC = 12, and CD = 13. (The diagram is not drawn to scale.)

x = 16.5

Solve for x. x = 21 x = 56 x = 22 x = 7

x = 21

Solve the problem below. x = 23; y = 246 x = 57; y = 246 x = 57; y = 123 x = 23; y = 123

x = 23; y = 123

Are the two figures similar? If so, give the similarity ratio. yes; 30 : 7 yes; 5 : 1 yes; 6 : 7 no

yes; 5 : 1

Corollary

you can have a maximum of 1 right of one botuse angle in a triangle.


संबंधित स्टडी सेट्स

POSC 100 - study guide #4 - Ch14

View Set

unit 2 - Cell Structures and Functions (pg. 46-60) ( pg.68-77)

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