Geometry 10
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
...
Alternate interior angles
...
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
...
Deductive Reasoning
...
Division Property
...
Exterior Angles
...
How to use the Properties of Iscosceles and Equilateral Triangles
...
Identifying Congruent Parts, Congruent Triangles and Writing Congruent Statements
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Identifying the types of Triangles
...
Inductive Reasoning
...
Interior Angles
...
Laws of Detachment
...
Laws of Logic
...
Laws of Syllogism
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Multiplication Property
...
Perpendicular Lines
...
Reflexive Property
...
Ruler Postulate
...
Same side interior angles
...
Segment Addition Postulate
...
Substitution Property
...
Subtraction Property
...
Symmetric Property
...
Transitive Property
...
Writing Proofs Using SSS, SAS, ASA, AAS, and HL plus CPCTC
...
acute angle
...
adjacent angles
...
angle
...
angle bisector
...
area
...
axiom
...
collinear points
...
compass
...
complementary angles
...
congruent angles
...
construction
...
coplanar
...
distance formula
...
intersect
...
line
...
midpoint formula
...
net
...
obtuse angle
...
opposite rays
...
parallel lines
...
parallel planes
...
perimeter
...
perpendicular bisector
...
perpendicular lines
...
plane
...
point
...
postulate
...
ray
...
right angle
...
segment
...
skew lines
...
space
...
straight angle
...
straightedge
...
supplementary angles
...
vertical angles
...
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.
