Honors Geometry Semester 1 Exam
m∠ABD = m∠DBC 5y - 3 = 3y + 15 2y = 18 y = 9 m∠DBC = 3(9) + 15 m∠DBC = 42°
BD bisects ∠ABC. m∠ABD = 5y - 3 and m∠DBC = 3y + 15. find m∠DBC
XY = ½(RS + QT) 2 x 16 = ½(RS + 20) x 2 32 = RS + 20 12 = RS
Find RS
sum of ext. ∠s = 360 measure of each ext. ∠ = 360/9 = 40°
Find the measure of each exterior angle of a regular nonagon
n = 35 sides sum of int. ∠s = (n-2)(180) = (35-2)(180) = (33)(180) = 5940 measure of each int. ∠ = 5940/35 = 169.7°
Find the measure of each interior angle of a regular 35-gon
∠CBD is an exterior angle n = 8 sides m∠CBD = 360/8 = 45°
Find the measure of ∠CBD in the regular octagon
HI = JK 2x - 10 = x + 80 x - 10 = 80 x = 90 HK = IJ 3y - 4 = y + 20 2y - 4 = 20 2y = 24 y = 12
Find x and y if HIJK is a parallelogram
BD ≅ AC = 16.3 CY + YA = 16.3 9.7 + YA = 16.3 YA = 6.6
Given isosceles trapezoid ABCD with AB ≅ CD, CY = 9.7 and BD = 16.3 Find YA.
ABCD is NOT a parallelogram because only one pair of opposite sides are congruent and only one pair of opposite angles are congruent. In a parallelogram, both pairs of opposite sides are congruent and both pairs of opposite angles are congruent.
If AB ≅ CD and ∠CBA ≅ ∠ADC, is ABCD a parallelogram? Justify your answer.
BA = AD 5x - 11 = 6x - 18 -11 = x - 18 7 = x BA = AD = BC 5(7) - 11 = 35 - 11 24 = BC
If ABCD is a rhombus, find BC
PQ = 5b + 6 PQ = 5(9) + 6 PQ = 45 + 6 = 51 SR = 8b - 21 SR = 8(9) -2 1 SR = 72 - 21 = 51 PQ = SR PS = 15a - 11 PS = 15(3) - 11 PS = 45 - 11 = 34 QR = 10a + 4 QR = 10(3) + 4 QR = 30 + 4 = 34 PS = QR Since PQ = SR and PS = QR, both pairs of opposite sides are congruent and PQRS is a parallelogram
If PQ = 5b + 6, QR = 10a + 4, RS = 8b-21, and SP = 15a-11, show that PQRS is a parallelogram for a = 3 and b = 9
JL = 556 / 2 = 278 278 - 67 = 211 miles
The map shows a linear sectioln of the highway. The Browns plan to drive 556 miles from Jackson to Fayetteville. They will stop to go to the bathroom in Little Rock which is at the midpoint of the trip. If they have already travelled 67 miles today, how much further must they travel before they stop in Little Rock for a bathroom break?
slope of BA → down 8, right 3 →⁸/₃ count down 8 and right 3 from C(5, 3) to get D(8, -5) BA ‖ CD slope of BC = 3-2/5+1 = ¹/₆ slope of AD = -5+6/8-2 = ¹/₆ BC ‖ AD
Three vertices of parallelogram ABCD are A(2, -6), B(-1, 2) and C(5, 3). Find the coordinates of vertex D.
mdpt. of PR = (1, -1) = (-3 + x/2, -2 + y/2) -3 + x/2 = 1 -3 + x = 2 x = 5 -1 = -2 + y/2 -2 = -2 + y y = 0 R = (5, 0) mdpt. of QS = (1, -1) = (-1 + x/2, 4 + y /2) 1 = -1 + x/2 2 + -1 + x 3 = x -1 = 4 + y/2 -2 = 4 + y -6 = y S = (3, -6)
Two vertices of a parallelogram are P(-3, -2) and Q(-1, 4) and the intersection of the diagonals is T(1, -1). Find the coordinates of the other two vertices, R and S.
diagonals of a rectangle are congruent KM = √(-5-3)² + (-1-1)² = √68 LN = √(-2-0)² + (4+4)² = √68 KM ≅ LN so KLMN is a rectangle diagonals of a rhombus are perpendicular slope of KM = -1-1/-5-3 = ¼ slope of LN = -4-4/0+2 = -4 KM ⊥ LN so KLMN is a rhombus since KLMN is a rectangle and a rhombus, it is also a square
Use the diagonals to determine whether a parallelogram with the vertices K(-5, -1), L(-2, 4), M(3, 1) and N(0, -4) is a reactangle, rhombus, or square. Give all names that apply.
m∠U ≅ m∠YZU m∠YZU = 52° m∠YZX + m∠YZU = 180° m∠YZX + 52° = 180° m∠YZX = 128° m∠YXZ + m∠XYZ + m∠YZX = 180° m∠YXZ + m∠XYZ + 128° = 180° m∠YXZ + m∠XYZ = 52° m∠YXZ = m∠XYZ m∠YXZ = m∠XYZ = ⁵²/₂ = 26° m∠ZXW = ½m∠YXZ = ½(26) = 13°
XW and YW are the angle bisectors of ∠YXZ and ∠XYZ, repectively. UY ≅ YZ ≅ XZ and m∠U = 52°. find m∠ZXW.
area of a triangle
a = 1/2bh
area of a rectangle
a = lw
area of a circle
a = πr²
∠1 ≅ ∠3 so ∠3 = 32.5 ∠3 + ∠4 = 90° 32.5 + ∠4 = 90° ∠4 = 57.5° ∠4 = ∠6 = 57.5° ∠4 + ∠5 + ∠6 = 180° 2(57.5°) + ∠5 = 180° 115 + ∠5 = 180° ∠5 = 65°
a billard ball bounces off the sides of a rectangular billards table in such a way that ∠1≅∠3, ∠4≅∠6, and ∠3 and ∠4 are complementary. if m∠1 = 32.5°, find m∠3, ∠4, and ∠5
transformation
a change in the position, size, or shape of a figure
inscribed
a circle in which each side of the polygon is tangent to the circle
polygon
a closed plane figure formed by three or more segments that intersect only at their endpoints
angle
a figure formed by two rays with a common endpoint
pentagon
a five-sided polygon
plane
a flat surface that has no thickness and extends forever
quadrilateral
a four-sided polygon
between
a given point B is _________ A and C if they all lie on the same line
perpendicular bisector
a line perpendicular to a segment at the segment's midpoint
transversal
a line that intersects two coplanar lines at two different points
auxiliary line
a line that is added to a figure to aid in a proof
parallel, half
a midsegment of a triangle is ________ to a side of a triangle and its length is _____ the length of that side (∆ midsegment thrm.)
nonagon
a nine-sided polygon
linear pair
a pair of adjacent angles whose non-common sides are opposite rays
ray
a part of a line that starts at an endpoint and extends forever in one direction
altitude of a triangle
a perpendicular segment from a vertex to the line containing the opposite side
coordinate plane
a plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
endpoint
a point at an end of a segment or the starting point of a ray
midpoint
a point that divides a segment into two congruent segments
point of concurrency
a point where three or more lines coincide
concave
a polygon that contains points in the exterior of the polygon
convex
a polygon that has no points in the exterior
regular polygon
a polygon that is both equilateral and equiangular
n-gon
a polygon with n sides
irregular polygon
a polygon with sides that are not all congruent or angles that are not all congruent
indirect proof
a proof in which the statement to be proved is assumed to be false and a contradiction is shown
triangle rigidity
a property of triangles that states that if the side lengths of a triangle are fixed, the triangle can have only one shape
trapezoid
a quadrilateral with exactly one pair of parallel sides
kite
a quadrilateral with exactly two pairs of congruent consecutive sides
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
parallelogram
a quadrilateral with two pairs of parallel sides
angle bisecter
a ray that divides an angle into two congruent angles
y = -4x + 1 slope = -4/1 y-int = (0, 1) x-int = 0 = -4x + 1 = -1 = -4x =1/4 distance = √(¼-0)² + (0-1)² = √(¼)² + (-1)² = √1.0623 ≈ 1.03
a right triangle is formed by the x-axis, the y-axis, and the line y = -4x + 1. find the length of the hypotenuse. round your answer to the nearest hundredth.
height
a segment from a vertex that forms a right angle with a line containing the base
diagonal
a segment that connects any two nonconsecutive vertices
midsegment of a triangle
a segment that joins the midpoints of two sides of the triangle
diameter
a segment that passes through the center of the circle and whose endpoints are on the circle
median of a triangle
a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side
radius
a segment whose endpoints are the center of the circle and a point on the circle
segment bisector
a segment, ray, line, or plane that intersects a segment at its midpoint
locus
a set of points that satisfies a given condition
heptagon
a seven-sided polygon
hexagon
a six-sided polygon
⁴⁴/₄₇₅ ≈ 0.0926 x 360° ≈ 33.5°
a spaceship flys completely around the moon in 475 days. determine the angle through which the spaceship travels over a period of 44 days.
conroe: y = x + 115 lakeville:-(y = 10x + 50) = 0 = -9x + 65 9x = 65 x = 7 conroe: 7 + 115 = $112 = 17mi lakeville: 10(7) + 50 = $120 = 17mi 55 + 17 = 72mi/h
a speeding ticket in conroe costs $115 for the first 10 mi/h over the speed limits and $1 for each additional mi/h. in lakeville, a ticket costs $50 for the first 10 mi/h over the speed limit and $10 for each additional mi/h. if the speed limit is 55 mi/h, at what speed will the tickers cost approx. the same?
postulate
a statement that is accepted as true without proof
line
a straight path that has no thickness and extends forever
divide ∆ into 2 30°-60°-90° ∆ 40 = hypotenuse 20 = short leg h = long leg h = 20√3 ≈ 34.64 inches yes, it will be big enough because the width and height are less than 46 and 42
a street sign has a shape of an equilateral triangle with side lengths og 40 inches. what is the height of the sign? will a rectangular metal sheet of 46 by 42 inches be big enough to make one sign?
coordinate proof
a style of proof that uses coordinate geometry and algebra
decagon
a ten sided polygon
corollary
a theorem whose proof follows directly from another theorem
triangle
a three-sided polygon
reflection
a transformation across a line
rotation
a transformation in which a figure is turned around a point
translation
a transformation in which all the points of a figure move the same distance in the same direction
congruent
a trapezoid is isosceles if and only if the disgonals are __________________ (thrm. 6-6-5)
isosceles trapezoid
a trapezoid with congruent legs
perimeter of 1 ∆ = 9(3) = 27in 120 / 27 ≈ 4.44 they can make 4 triangle instruments
a triangle is a percussion instrument made by steel rod. each triangle instrument is an equilateral triangle. each side of a triangle is 9 inches long. how many triangle instruments can be made from a piece of steel rod that is 120 inches long?
isosceles triangle
a triangle with at least two congruent sides
scalene triangle
a triangle with no congruent sides
obtuse triangle
a triangle with one obtuse angle
right triangle
a triangle with one right angle
acute triangle
a triangle with three acute angles
equiangular triangle
a triangle with three congruent angles
equilateral triangle
a triangle with three congruent sides
dodecagon
a twelve-sided polygon
construction
a way of creating a figure that is more precise
right
all _______ angles are congruent (right ∠ congruence thrm.)
right angle
an angle that measures 90 degrees
acute angle
an angle that measures greater than 0 degrees but less than 90 degrees
obtuse angle
an angle that measures more than 90 degrees but less than 180 degrees
x = 12 + ½(180 - x) x = 12 + 90 - ½x x = 102 -½x 1.5x = 102 x = 68 angle = 68°
an angle's measure is 12 degrees more than ½ the measrue of its supplement. find the measure of the angle.
x = 6 + 3(90-x) x = 276 - 3x 4x = 276 x = 69°
an angle's measure is 6 degrees more than 3 times the measure of its complement. find the measure of the angle
octagon
an eight-sided polygon
remote interior angle
an interior angle that is not adjacent to the exterior angle
3x + 6 + 3x + 6 + 7x = 64 13x + 12 = 64 13x = 52 x = 4
an isosceles triangle has a perimeter of 64 cm. the congruent sides measure (3x + 6)cm. the length of the third side is 7x cm. what is the value of x?
corresponding angles
angles in the same position in polygons with an equal number of sides
congruent angles
angles that have the same measure
base
any side of a triangle
parallel lines
are coplanar and do not intersect
undefined terms
cannot be defined by using other figures; point, line, and plane
a) scalene b) isosceles c) equilateral d) scalene e) isosceles
classify the triangles by their side lengths
∆ABC = acute ∆ ∆BCD = obtuse ∆ ∆ABD = obtuse ∆
classify ∆ABC, ∆BCD, and ∆ABD by angle measures of each triangle
m∠CAB > m∠DAB because ∠CAB is opposite the longer side
compare m∠CAB and m∠DAB
CPCTC
corresponding parts of congruent triangles are congruent
m∠CAB + m∠ABC + m∠BCA = 180° 54° + 27° + m∠BCA = 180° 81° + m∠BCA = 180° m∠BCA = 99° m∠BCA + m∠BCD + m∠ECD = 180° 99° + 39° + m∠ECD = 180° 138°+ m∠ECD = 180° m∠ECD = 42°
daphne folded a triangular sheet of paper into the shape shown. find m∠ECD given that m∠CAB = 54°, m∠ABC = 27°, and m∠BCD = 39°
a) AC ≅ EC (given) ∠ACB ≅ ∠ECD (vert. ∠s thrm.) BC ≅ DC (given) ∆ACB ≅ ∆ECD (SAS) b) ∠ABD ≅ ∠CBD (given) ∠ADB ≅ ∠CDB (given) BD ≅ BD (reflex. prop. ≅) ∆ADB ≅ ∆CDB (ASA) c) ∠BAP ≅ ∠CDP (right ∠s thrm) AP ≅ DP (given) ∠APB ≅ ∠DPC (vert. ∠s thrm) ∆APB ≅ ∆DPC (ASA)
determine if you can use SSS, SAS, ASA, AAS, or HL to prove the following pairs of triangle congruent. if so, explain why using correct notation and listing the pairs of corresponding congruent parts of the trangles.
6x - 12y = -24 -12y = -6x - 24 y = 1/2x + 2 3y = 2x + 18 y = 2/3x + 6 lines intersect
determine whether the lines are parallel, intersect, or coincide. 6x - 12y = -24, 3y = 2x + 18
distance formula
distance = √(x₂ - x₁)² + (y₂ - y₁)²
length
distance between two points
bisects
divides
see picture
draw and label a pair of opposite rays ←AB and →BC
side of a polygon
each segment that forms a polygon
∆1 hyp = 5 × √2 = 5√2 ∆2 legs = 5√2 ∆2 hyp. = 5√2 × √2 = 5 × 2 = 10 ∆3 hyp. = 10/2 = 5 x (∆3 legs) = 5 / √2 × √2 / √2 x = 5√2 / 2
each triangle is a 45°-45°-90° ∆. find the value of x
circumscribed
every vertex of the polygon lies on the circle
AB = CB 4y + 12 = 5y - 6 12 = y - 6 18 = y BC = 5(18) - 6 = 90 - 6 = 84
find BC
∠B = ∠G because ∆BUG is isosceles ∠B = x + 10 = ∠G m∠B + m∠U + m∠G = 180° x + 10 + 3x + x + 10 = 180° 5x + 20 = 180 5x = 160 x = 32 m∠G = (x + 10)° = 32 + 10 = 42°
find m∠G
the ⊥ bisector of AC is y = 0.5 the ⊥ bisector of CB is x = -1 the intersection of the circumcenter is (-1, 0.5)
find the circumcenter of ∆ABC with vertices A(-3, 3), B(1, -2), and C(-3, -2)
C = 2πr C = 2π(8) C = 16(π) C ≈ 50.2ft A = πr² A = π(8)² A = 64(π) A ≈ 201ft²
find the circumference and area of the circle. use 3.14 for π and round your answer to the nearest tenth
midpoint = [4+(-4) / 2, -6+2 / 2] midpoint = [0/2, -4/2] midpoint = (0, -2)
find the coordinates of the midpoint of AB with endpoints A(4, -6) and B(-4, 2)
DE = |-3-(-9)| DE = |6| = 6
find the length of DE
top half of ∠1 is 25° because it corresponds with the 25° angle in the picture bottom half of ∠1 is 40° because it is a same-side int. angle with the 140° angle and same side int. ∠s are supplementary m∠1 = 25° + 40° = 65°
find the measure of angle 1 in the diagram (hint: draw a line parallel to the given parallel lines)
m∠C = m∠F 3x² = 7x² - 16 -4x² = -16 x² = 4 √x² = √4 x = 2 m∠C = 3(2)² m∠C = 3(4) = 12° m∠F = 7(2)² - 16 m∠F = 7(4) - 16 m∠F = 28 - 16 = 8°
find the measure of angle C and angle F in the triangles given that ∠A = ∠E and ∠C = 3x² and ∠F = 7x² - 16
m∠2 + 122° = 180° m∠2 = 58° m∠3 = m∠2 = 58° because ∠2 and ∠3 are alt. int. ∠s m∠1 = 180° - 58°- 58° = 64°
find the measure of each numbered angle
complement of ∠B = 90 - (6x - 5) complement of ∠B = (95 - 6x)° supplement of ∠B = 180 - (6x - 5) supplement of ∠B = (185 - 6x)°
find the measure of the complement and the supplement of ∠B if ∠B = 6x - 5.
m∠ACB = 37° m∠DBC = 37° because ≅ to ∠ACB
find the measure of ∠DBC given that ∠A≅∠D, ∠ABC≅∠DCB, and m∠ACB = 37°
MP bisects NO and is the same distance away from the endpoints N and O MO = MN = 7 NP = PO = 3 NO = NP + PO = 3 + 3 = 6
find the measures of MN and NO
m∠A + m∠B = m∠ACD 5y + 3 + 4y + 8 = 146 9y + 11 = 146 9y = 135 y = 15 m∠A = 5(15) + 3 = 78°
find the measures of ∠A and ∠B in ∆ABC
16² + x² = 20² 256 + x² = 400 x² = 144 √x² = √144 x = 12 yes, they are a pythagorean triple because they satisfy a² + b² = c² and they are all nonzero whole numbers
find the missing side length. tell if the side lengths form a pythagorean triple. explain.
8² + 15² = x² 64 + 225 = x² 289 = x² √289 = √x² 17 = x
find the missing side of the triangle
4x + 1 = 5x - 0.5 1 = x - 0.5 1.5 = x AC = 4(1.5) + 1 = 6 + 1 = 7 BC = 5(1.5) - 0.5 = 7.5 - 0.5 = 7 AB = 9(1.5) - 1 = 13.5 - 1 = 12.5
find the three side lengths of the triangle
midsegment is ½ the side that is opposite from it midsegment = 69 69 = ½(n² - 8) 138 = n² - 8 146 = n² √146 = √n² n ≈ 12.08
find the value of n in the triangle
2x + 5 = 8 - 4x 6x = 3 x = ½
find the value of x
p = 64cm equilateral ∆s are also equiangular 5 - 11x + 5 - 11x + 5 - 11x = 180° 15 - 33x = 180° -33x = 165° x = -5
find the value of x
30°-60°-90° ∆ x = short leg y = long leg 14 = hypotenuse x = ¹⁴/₂ = 7in. y = 7 × √3 = 7√3in.
find the value of x and y. express your answer in simplest radical form.
45°-45°-90° ∆ x = leg 7 = hypotenuse x = 7 / √2 × √2 / √2 = 7√2 / 2 ft
find the value of x. express your answer in simplest radical form
5x + 55 = 2x + 100 3x = 45 x = 15 5(15) + 55 = 75 + 55 = 130° 2(15) + 100 = 30 + 100 = 130° by the corresponding ∠s post.
find x and the measures of the 2 angles in the picture if the horizontal lines are parallel. what theorem or postulate allows us to find the angle measures?
17x + 14 + 4x - 2 = 180° 21x + 12 = 180° 21x = 168° x = 8 17(8) + 14 = 136 + 14 = 150° 4(8) - 2 = 32 - 2 = 30° by the same-side int. ∠s thrm.
find x and the measures of the 2 angles in the picture if ℓ and m are parallel. what theorem or postulate allows us to find the measures of the angles?
3y + 53 = 7y - 55 108 = 4y y = 27 3(27) + 53 = 81 + 53 = 134° 7(27) - 55 = 189 - 55 = 134° by the alt. ext. ∠s thrm.
find y and the measures of the 2 angles in the picture. what theorem or postulate allows us to find the measures of the angles?
BA ≅ ED (given → hypotenuse) AC ≅ DF (given → leg) ∆ABC ≅ ∆DEF by HL
for these triangles, which postulate or theorem makes them congruent, and what is the traingle congruence statement?
straight angle
formed by two opposite rays and measures 180 degrees
0 to 180
given AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from ___ to ___ (protractor post.)
AC ≅ BD
given collinear points A, B, C, and D arranged as shown, if AB ≅ CD then ___ ≅ ___ (common segments thrm.)
measure
given in degrees
∆ABC ≅ ∆EDC m∠B = 63° m∠ABC = 90° m∠BAC = 180° - 63° - 90° = 27° ∠BAC corresponds to ∠DEC and since ∆ABC ≅ ∆EDC, corresponding parts are congruent m∠DEC = m∠BAC = 27°
given that ∆ABC ≅ ∆EDC and m∠B = 63°, find the measure of ∠DEC
ME ≅ RE because ME = RE = 8cm MT ≅ RP because MT = RP = 7cm TE ≅ PE because TE = PE by def. of midpoint therefore, ∆MTE ≅ ∆RPE by SSS
given the lengths marked on the figure and that E is the midpoint of MR, use SSS to explain why ∆MTE ≅ ∆RPE. be specific and use correct notation!
PQ = ½BC PQ = ½(6) = 3
given ∆ABC with AB = 18 and BC = 6, find the length of midsegment PQ
statements: 1. ∠M ≅∠P 2. ∠LNM ≅ ∠ONP 3. ∠MLN ≅ ∠PON 4. ML ≅ PO 5. LO bisects MP, MP bisects LO 6. LN ≅ ON, MN ≅ PN 7. ∆LMN ≅ ∆OPN reasons: 1. given 2. vert. ∠s are ≅ 3. third ∠s thrm. 4. given 5. given 6. def. of bisector 7. def. of ≅ ∆s
given: LO bisects MP, MP bisects LO, ML ≅ PO, ∠M ≅∠P prove: ∆LMN ≅ ∆OPN (7 steps)
right
if one angle of a parallelogram is a _________ angle, then the parallelogram is a rectangle(thrm. 6-5-1)
opposite angles
if one diagonal of a parallelogram bisects a pair of ____________________, then the parallelogram is a rhombus (thrm. 6-5-5)
opposite sides
if one pair of __________________ of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram (thrm. 6-3-1)
congruent
if one pair of consecutive sides of a parallelogram is _________________, then the parallelogram is a rhombus (thrm. 6-5-3)
diagonals
if the _____________ of a parallelogram are perpendicular, then the parallelogram is a rhombus (thrm. 6-5-4)
diagonals
if the _________________ of a quadrilateral bisect each other, then the quadrilateral is a parallelogram (thrm. 6-3-5)
congruent
if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are _________ (HL congruence)
right
if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a ______ triangle (conv. of the pythagorean thrm.)
congruent
if three sides of one triangle are congruent to three sides of another triangle then the triangles are ________ (SSS congruence)
planes
if two _____ intersect, then they intersect in exactly one line (post. 1-1-5)
lines
if two _____ intersect, then they intersect in exactly one point (post. 1-1-4)
congruent
if two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are ________ (AAS congruence)
congruent
if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are ___________ (ASA congruence)
supplementary
if two angles are ___________ to the same angle (or to two congruent angles), then the two angles are congruent (congruent supplements thrm.)
complementary
if two angles are ____________ to the same angle (or two congruent angles), then the two angles are congruent (congruent complements thrm.)
supplementary
if two angles form a linear pair then they are ____________ (linear pair thrm.)
congruent
if two angles of a traignle are congruent, then the sides opposite those angles are __________ (conv. of isosceles ∆ thrm.)
opposite
if two angles of a triangle are not congruent, then the longer side is _________ the larger angle (thrm. 5-5-2)
congruent
if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are _________ (third ∠s thrm.)
right
if two congruent angles are supplementary, then each angle is a _______ angle (thrm. 2-7-3)
parallel
if two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are ________ (conv. of alternate exterior ∠s post.)
parallel
if two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are ________ (conv. of alternate interior ∠s post.)
parallel
if two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are ________ (conv. of corresponding ∠s post.)
parallel
if two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are ________ (conv. of corresponding ∠s post.)
parallel
if two coplanar lines are perpendicular to the same line, then the two lines are ________ (thrm. 3-4-3)
perpendicular
if two intersecting lines form a linear pair of congruent angles, then the lines are ____________ (thrm. 3-4-1)
congruent
if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are ________ (alternate exterior ∠s post.)
congruent
if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are ________ (alternate interior ∠s post.)
congruent
if two parallel lines are cut by a transversal, then the pairs of corresponding angles are ________ (corresponding ∠s post.)
supplementary
if two parallel lines are cut by a transversal, then the pairs of same-side interior angles are ________ (same-side interior ∠s post.)
plane
if two points lie in a plane, then the line containing those points lies in the ______ (post. 1-1-3)
congruent
if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are _________ (SAS congruence)
congruent
if two sides of a triangle are congruent, then the angles opposite the sides are __________ (isosceles ∆ thrm.)
opposite
if two sides of a triangle are not congruent, then the larger angle is _________ the longer side (thrm. 5-5-1)
across from
if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the larger included angle is ________ the longer third side (conv. of hinge thrm.)
across from
if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is ________ the larger included angle (hinge thrm.)
2 × the length of the shorter leg, the length of the shorter leg × √3
in a 30°-60°-90° triangle, the length of the hypotenuse is ____________ and the length of the longer leg is ___________ (30°-60°-90° ∆ thrm.)
congruent, the length of the leg × √2
in a 45°-45°-90° triangle, both legs are ________, and the length of the hypotenuse is ____________ (45°-45°-90° ∆ thrm.)
slope
in a coordinate plane, two nonvertical lines are parallel if and only if they have the same _______ (parallel lines thrm.)
-1
in a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is ___ (perpendicular lines thrm.)
perpendicular
in a plane, if a transversal is perpendicular to one of the two parallel lines, then it is ____________ to the other line (perpendicular transversal thrm.)
legs, hypotenuse
in a right triangle, the sum of the squares of the lengths of the _____ is equal to the sum of the squares of the length of the ______ (pythagorean thrm.)
m∠1 = 17x + 9 = 17(3) + 9 = 51 + 9 = 60° m∠2 = 14x + 18 = 14(3) + 18 = 42 + 18 = 60° m∠1 = m∠2 ∠1 ≅ ∠2 l ‖ m by conv. of alt. int. ∠s thrm.
in a swimming pool, two lanes are represented by lines l and m. if a string of flags strung across the lanes is represented by transversal t, and x = 3, show that the lanes are parallel
longest, obtuse, acute
in ∆ABC, c is the length of the ______ side. if c² > a² + b², then ∆ABC is an ______ triangle. if c² < a² + b², then ∆ABC is an _____ triangle (pyhagorean inequality thrm.)
slope of DE = -1 - 2 / 0 = 2 = -3 / -2 = ³/₂ slope of AC = -6 -0 / 1 - 5 = -6 / -4 = ³/₂ slopes are the same so DE ‖ AC DE = √(2 - 0)² + (2 + 1)² = √4 + 9 = √13 AC = √(5 - 1)² + (0 + 6)² = √16 + 36 = √52 = √4 x 13 = 2√13 √13 is ½ of 2√13 so DE = ½AC
in ∆ABC, show that midsegment DE is parallel to AC and that DE = ½AC
perpendicular lines
intersect at 90° angles
LK is the midsegment LK = ½MN 220 = ½MN MN = 440m
john wants to measure the width of a reservoir. he measures a triangle at one side of the reservoir as shown in the diagram. what is the width of the reservoir (MN across the base)?
∠IJL ∠LJK ∠IJK
name all of the angles in the diagram
corresponding angles: ∠A and ∠E ∠B and ∠F ∠D and ∠H ∠C and ∠G alternate interior angles: ∠B and ∠H ∠C and ∠E alternatre exterior angles ∠A and ∠G ∠D and ∠F same-side interior angles ∠B and ∠E ∠C and ∠H
name all the pairs of corresponding, alternate interior, alternate exterior, and same-side interior angles
∠LMN and ∠PMO ∠OMN and ∠PML
name all the pairs of vertical angles
A, C, R
name three collinear points
line AZ and line BX
name two lines in the figure
point
names a location and has no size
alternate interior angles
nonadjacent angles that lie on opposite sides of the traversal between lines
skew lines
not coplanar, not parallel, and do not intersect
congruent
of the diagonals of a parallelogram are ________________, then the parallelogram is a rectangle(thrm. 6-5-2)
base angle of a trapezoid
one of a pair of consecutive angles whose common side is a base of the trapezoid
90° - 42.7° = 47.3°
one of the acute angles in a right triangle has a measure of 42.7°. what is the measure of the other acute angle?
leg of a trapezoid
one of the two nonparallel sides of the trapezoid
base of a trapezoid
one of the two parallel sides of the trapezoid
parallel planes
planes that do not intersect
PC = ²/₃PY by centroid thrm. PC = ²/₃(4.2) = 2.8
point C is the centroid of ∆PQR, PY = 4.2, and QC = 2. find PC.
collinear
points that lie on the same line
coplanar
points that lie on the same plane
congruent segments
segments that have the same length
corresponding sides
sides in the same position in two different polygons that have the same number of sides
slope formula
slope = y₂ - y₁ / x₂ - x₁
8 + 12 = 20 > 17 8 + 17 = 25 > 20 12 + 17 = 29 > 8 yes, they can be a triangle 8² + 12² = 17² 64 + 144 = 289 208 < 289 a² + b² < c² so it would be an obtuse triangle
tell if the measures 8, 12, and 17 can be the side lengths of a traingle. if so, classify the triangle as acute, right, or obtuse
14 + 15 = 29 <30 no
tell whether a traignle can have sides with lengths 14, 15, and 30
6 + 8 = 14 > 10 6 + 10 = 16 > 8 8 + 10 = 18 > 6 yes
tell whether a traingle can have lengths 6, 8, and 10
distance
the absolute value of the difference of the coordinates
complementary
the acute angles of a right triangle are ______________ (cor. 4-2-2)
vertex angle
the angle formed by the legs of an isosceles triangle
included angle
the angle formed by two adjacent sides of a polygon
²/₃
the centroid of a triangle is located ___ of the distance from each vertex to the midpoint of the opposite side (centroid thrm.)
vertices
the circumcenter of a triangle is equidistant from the _______ of the trangle (circumcenter thrm.)
vertex
the common endpoint of an angle
vertex of a polygon
the common endpoint of two sides
included side
the common side of two consecutive angles of a polygon
midsegment of a trapezoid
the segment whose endpoints are the midpoints of the legs of the trapezoid
interior of an angle
the set of all points between the sides of the angle
interior
the set of all points inside the figure
exterior of the angle
the set of all points outside the angle
exterior
the set of all points outside the figure
hypotenuse
the side of a right triangle opposite the right angle
base
the side opposite the vertex angle
w = 3x, h = 2x (3x)² + (2x)² = 32² 9x² + 4x² = 1024 13x² = 1024 x² ≈ 78.77 √x² ≈ √78.77 x ≈ 8.88 width = 3(8.88) = 26.64 height = 2(8.88) = 17.76
the size of a tv screen is given by the length of its diagonal. the screen aspect ratio is the ratio of its width to its height. the screen asprect ratio of a standard tv screen is 3:2. what are the width and height of a 32" tv screen?
greater
the sum of any two side lengths of a triangle is ________ than the third side length (∆ inequality thrm.)
180°
the sum of the angle measures of a triangle is ____ (triangle sum thrm.)
360°
the sum of the exterior angle measure, one angle at each vertex, of a convex polygon is _____ (polygon exterior ∠ thrm.)
(n - 2)180°
the sum of the interior angle measures of a convex polygon with n sides is _______ (polygon ∠ sum thrm.)
perimeter
the sum of the side lengths of the figure
legs of an isosceles triangle
the two congruent sides of an isosceles triangle
legs
the two sides of a right triangle that meet to form the right angle
concurrent
three or more lines that intersect at one point
plane
through any three noncollinear points there is exactly one ______ containing them (post. 1-1-2)
line
through any two points there is exactly one ______ (post. 1-1-1)
parallel
through point P not on line ℓ, there is exactly one line ________ to ℓ (parallel post.)
base angles
two angles that have the base as a side
adjacent angles
two angles that share a common vertex and side, but have no common interior points
supplementary angles
two angles whose measures have a sum of 180 degrees
complementary angles
two angles whose measures have a sum of 90 degrees
m∠Y = 180° - 120° = 60° m∠X = m∠Y = 60° m∠Z = 180° - m∠X - m∠Y = 180° - 60° - 60° = 60°
two cities on a map represented by points X and Y are equidistant from another city represented by point Z. what is the m∠Z?
verticle angles
two nonadjacent angles formed by two intersecting lines
congruent polygons
two polygons whose corresponding sides and angles are congruent
opposite rays
two rays that have a common endpoint and form a line
p = 39 each side = 39 / 3 = 13 3x - 5 = 13 3x = 18 x = 6 x² - 23 = 13 x² = 36 √x² = √36 x = 6
two sides of an equilateral traignle measure (3x - 5) units and (x² - 23) units. if the perimeters of the traignle is 39 units, what is the value of x?
slope of LM = 5 - 2 / 2 -(-2) = 3/4 slope of NP = -2 - 2 / 3 - 0 = -4/3 slopes are opposite reciprocals so LM is perpendicular to NP
use slopes to determine whether the lines are parallel, perpendicular, or neither. LM and NP for L(-2, 2), M(2, 5), N(0, 2) and P(3, -2)
∠1 ≅ ∠2 because they are corresponding ∠s ℓ ‖ m by the conv. of corresponding ∠s post.
use the converse of the corresponding ∠s post. and ∠1 ≅∠2 to show that ℓ ‖ m
distance formula: d = √(-4-2)² + (8-(-1))² d = √(-6)² + (9)² d = √36 + 81 d = √117 ≈ 10.8 pythagorean theorem: units down from W(-4, 8) to V(2, -1) = 9 = a units across from W(-4, 8) to V(2, -1) = 6 = b 9² + 6² = c² 81 + 36 = c² 117 = c² √117 = c ≈ 10.8
use the distance formula and the pythagorean theorem to find the distance, to the nearest tenth, from V(2, -1) to W(-4, 8)
m∠1 = 2x = 2(50) = 100° m∠2 = x + 50 = 50 + 50 = 100° m∠1 = m∠2 ∠1 ≅ ∠2 ℓ ‖ m by con. of alt. int. ∠s thrm.
use the information m∠1 = 2x°, m∠2 = x + 50°, x = 50 and the theorems you have learned to show that ℓ ‖ m
m = 4 - 1 / 5 - 1 = 2/4 = 1/2
use the slope formula to find the slope of the line
m = 1 - 2 / 4 - (-3) m = -¹/₇
use the slope formula to find the slope of the line through (-3, 2) and (4, 1)
congruent
verticle angles are ______ (verticle ∠s thrm.)
AK ≅ AK by reflex. prop. of ≅ JK ≅ NK is given you need to know that the included angles, ∠JKA and ∠NKA are congruent to say that ∆JKA ≅ ∆NKA by SAS
what additional information do you need to prove ∆JKA ≅ ∆NKA by the SAS post.? be specific and use correct notation!
adjacent= ∠1 and ∠2 ∠2 and ∠3 ∠4 and ∠CAG linear pair= ∠FAC and ∠GAC ∠FAB and ∠BAG
which pairs of angles are adjacent? which form linear pairs?
(x, y) → (x + 6, y - 2)
write a rule for the translation from the left triangle to the right triangle
midpoint of AB = (-4 + 8 / 2, 1 + 7 / 2) = (⁴/₂, ⁸/₂) = (2, 4) slope of AB = 7 - 1 / 8 - (-4) = ⁶/₁₂ = ¹/₂ slope of ⊥ bisector = -2 write equation in point-slope form y - 4 = -2 (x - 2)
write an equation in point-slope form of the perpendicular bisector of the aegment with the endpoints A(-4, 1) and B(8, 7)
RS > RV 14 > x + 5 9 > x
write and solve an inequality for x
y + 2 = ³/₅ (x - 5) y + 2 = ³/₅x - 3 y = ³/₅x - 5
write the equation of the line through (5, -2) with slope ³/₅ in slope-intercept form
y - 2 = 6(x + 1)
write the equation of the line which slope 6 through the point (-1, 2) in point-slope form
AC, BC, AB
write the sides of ∆ABC in order from shortest to longest
point-slope form
y - y₁ = m(x - x₁)
slope-intercept form
y = mx + b
image
the resulting figure after a transformation
equidistant
the same distance from two or more objects
degree
1/360 of a circle
DC = ½(CA) DC = ½(24) DC = 12 inches
A flag design is rectangular with stripes that go from the cornet to the center of the flag. The dimensions of the falg are DC = 18 inches and CA = 24 inches. Find MB
pythagorean triple
A set of three nonzero whole numbers a, b, and c such that a² + b² = c²
Opposite sides are congruent so ABCD is a parallelogram. Diagonals are congruent so it is a rectangle. A pair of consecutive sides is congruent so ABCD is a square.
Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: AB ≅ CD, BC ≅ AD, AD ≅ DC, AC ≅ BD Conclusion: ABCD is a square
diagonals of a kite are perpendicular, so all triangles are right triangles m∠SOP = 90° m∠SPO + m∠PSO = 90° 46° + m∠PSO = 90° m∠PSO = 44° m∠PSO + m∠RSO = m∠PSR 44° + 65° = 109° = m∠PSR m∠PSR = m∠PQR = 109° m∠ROS = 90° m∠RSO + m∠SRO = 90° 65° + m∠SRO = 90° m∠SRO = 25°
In kite PQRS, m∠SPO = 46° and m∠RSO = 65°. Find m∠PQR
NO = LM = 12.1 LO = MN = 13.7 P = 2(12.1) + 2(13.7) P = 51.6
In parallelogram LMNO, NO = 12.1 and LO = 13.7. What is the perimeter of parallelogram LMNO?
PQRS is always a parallelogram because as you raise or lower the binoculares the angles change but PQ = RS and QR = SP no mater what, whcih means both pairs of opposide sides are congruent.
In the parallelogram mount, there are bolts at P, Q, R, and S, such that PQ = RS and QR = SP. The frame PQRS moves when you raise or lower the binoculars. Why is PQRS always a parallelogram?
HJ = JK 3x + 5 = 9x - 3 8 = 6x x = 1¹/₃ HJ = 3(1¹/₃) + 5 HJ = 4 + 5 = 9 JK = 9(1¹/₃) - 3 JK = 12 - 3 = 9 HK = 9 + 9 = 18
J is the midpoint of HK. HJ = 3x + 5 and JK = 9x - 3. find HJ, JK, and HK.
AX = XC = 4in AB = BC = 5in AD = CD = 10in (AX)² + (BX)² = (AB)² 4² + BX² = 5² 16 + (BX)² = 25 √BX² = √9 BX = 3 (AX)² + (XD)² = (AD)² 4² + XD² = 10² 16 + XD² = 100 √XD² = √84 XD = 9.17 BX + XD + AC = 3 + 9.17 + 8 = 20.17 inches 20.17/7 ≈ 2.88 Joes needs 3 packages
Joe is designing a kite and wants to create a design connecting opposite corners from point B to point D and from point A to point C. Find the amount of cording needed to make the kite. One package of cording contains 7 inches of cord. How many packages does Joe need?
(0, 1) = [-3+x/2, -1+y/2] -3 + x / 2 = 0 -3 + x = 0 x = 3 -1 + y / 2 = 1 -1 + y = 2 y = 3 N = (3,3)
M is the midpoint of LN. L has the coordinates (-3, -1) and M has the coordinates (0, 1). find the coordinates of N
P is the circumcenter of ∆ABC so it's the same distance from the vertices of ∆ABC AP = CP = BP = 3.6
MP, NP, and QP are the perpendicular bisectors of ∆ABC. find CP and BP.
MP = MN + NP 5y + 9 = 17 + 3y 2y + 9 = 17 2y = 8 y = 4 MP = 5y + 9 MP = 5(4) + 9 MP = 29
N is between M and P. find MP
QS = RT 7a - 0.5 = 5a + 2.3 2a = 2.8 a = 1.4
QS = 7a - 0.5 and RT = 5a + 2.3. Find the value of a so that QRST is isosceles.
AB = √(2-1)² + (-1-2)² = √10 BC = √(5-2)² + (0+1)² = √10 CD = √(5-4)² + (0-3)² = √10 DA = √(4-1)² + (3-2)² = √10 all sides are congruent AB = -1-2/2-1 = -3 BC = -1-0/2-5 = 1/3 slopes are opposite reciprocals so AB ⊥ BC
Show that all four sides of square ABCD are congruent and that AB ⊥ BC
slope of EF = 8-5/2+1 = 1 slope of HG = 4-1/4-1 = 1 EF ‖ HG slope of EH = 5-1/-1-1 = -2 slope of FG = 8-4/2-4 = -2 EH ‖ FG both pairs of opposite sides are parallel so EFGH is a parallelogram
Show that quadrilateral EFGH is a parallelogram
ABCD would be a rectanlge because opposite angles are congruent making it a parallelogram and all 4 angles are 90° making it a rectangle
The side of a cardboard box is a quadrilateral with ∠ADC ≅ ∠ABC and ∠BCD ≅ ∠BAD. If m∠ADC = 90°, what is the most accurate description of ABCD?
circumference of a circle
c = 2πr or c = πd
statements: 1. P is the midpoint of QT and SR 2. QP = PT, SP = PR 3. QP ≅ PT, SP ≅ PR 4. ∠QPR ≅ ∠TPS 5. ∆OPR ≅ ∆TPS reasons: 1. given 2. def. of midpoint 3. def. of ≅ segments 4. vert. ∠s thrm. 5. SAS
given: P is the midpoint of OT and SR prove: ∆QPR ≅ ∆TPS (5 steps)
statements: 1. p ‖ r 2. ∠3 ≅ ∠2 3. ∠1 ≅ ∠3 4. ∠1 ≅ ∠2 5. ℓ ‖ m reasons: 1. given 2. alt. ext. ∠s thrm. 3. given 4. trans. prop. of ≅ 5. conv. of corr. ∠s post.
given: p ‖ r, ∠1 ≅ ∠3 prove: ℓ ‖ m (5 steps total)
3 pairs of ≅ sides: AB ≅ PQ BC ≅ QR CA ≅ RP 3 pairs of ≅ angles: ∠A ≅ ∠P ∠B ≅ ∠Q ∠C ≅ ∠R
given: ∆ABC ≅ ∆PQR identify all pairs of congruent corresponding parts. use correction notation!
statements: 1. ∠ABC ≅ ∠CBE 2. CB ⊥ AF 3. DE ⊥ AF 4. CB ‖ DE reasons: 1. given 2. 2 intersecting lines form lin. pair of ≅ ∠s then lines ⊥ 3. given 4. 2 lines ⊥ to same lines then 2 lines ‖
given: ∠ABC ≅ ∠CBE, DE ⊥ AF prove: CB ‖ DE (4 steps)
see picture
graph the line y + 1 = -3(x - 5)
parallel: AC and BD perpendicular: BD and DE skew: EF and AC
identify a pair of parallel, perpendicular, and skew segments
reflection about the x-axis ∆EFG → ∆E'F'G'
identify the transformation. then use arrow notation to describe the transformation.
a) transveral k same-side int. ∠s b) transversal k same-side int. ∠s c) transversal j alt. int. ∠s
identify the transversal and classify each angle pair. a) ∠1 and ∠3 b) ∠1 and ∠2 c) ∠6 and ∠5
between
if B is _______ A and C, then AB + BC = AC (seg. addition post.)
CP bisects ∠ACB so it's the same distance away from the sides of the angle BP = 3.7 AP = BP = 3.7
if CP bisects ∠ACB and BP = 3.7, find AP
m∠PQR
if S is in the interior of ∠PQR, then m∠PQS + m∠SQR = ________ (∠ addition post.)
bisector
if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the ________ of the angle (conv. of the ∠ bisector thrm.)
perpendicular bisector
if a point is equidistant from the endpoints of a segment, then it is on the ___________________ of the segment (conv. of perpendicular bisector thrm.)
equidistant
if a point is on the bisector of an angle, then it is __________ from the sides of the angle (∠ bisector thrm.)
equidistant
if a point is on the perpendicular bisector of a segment, then it is _________ from the endpoints of the segment (perpendicular bisector thrm.)
opposite angles
if a quadrilateral is a kite, then exactly one pair of __________________ is congruent (thrm. 6-6-2)
perpendicular
if a quadrilateral is a kite, then its diagonals are __________________ (thrm. 6-6-1)
supplementary
if a quadrilateral is a parallelogram, then its consecutive angles are _________ (thrm. 6-2-3
bisect
if a quadrilateral is a parallelogram, then its diagonals _______ each other (thrm. 6-2-4)
congruent
if a quadrilateral is a parallelogram, then its opposite angles are ________ (thrm. 6-2-2)
congruent
if a quadrilateral is a parallelogram, then its opposite sides are ________ (thrm. 6-2-1)
parallelogram
if a quadrilateral is a rectangle, then it is a __________________ (thrm. 6-4-1)
congruent
if a quadrilateral is a rectangle, then its diagonals are _________________ (thrm. 6-4-2)
opposite angles
if a quadrilateral is a rhombus, then each diagonal bisects a pair of _______________________ (thrm. 6-4-5)
perpendicular
if a quadrilateral is a rhombus, then its diagonals are ___________ (thrm. 6-4-4)
base angles
if a quadrilateral is an isosceles trapezoid, then each pair of __________________ are congruent (thrm. 6-6-3)
isosceles
if a trapezoid has one pair of congruent base angles, then the trapezoid is _________________ (thrm. 6-6-4)
equilateral
if a triangle is equiangular, then it is _________ (cor. 4-8-4)
equiangular
if a triangle is equilateral, then it is _________ (cor. 4-8-3)
supplementary
if an angle of a quadrilateral is ____________________ to both of its consecutive angles, then the quadrilateral is a parallelogram (thrm. 6-3-4)
opposite sides
if both pairs of ____________________ of a quadrilateral are congruent, then the quadrilateral is a parallelogram (thrm. 6-3-2)
opposite angles
if both pairs of ______________________ of a quadrilateral are congruent, then the quadrilateral is a parallelogram (thrm. 6-3-3)
see picture for graph slope = 455 - 260 / 7 - 4 slope = 195 / 3 = 65 justin's average speed is 65 mi/h
justin is driving from home to his college dormitory. at 4:00 pm, he is 260 miles from home. at 7:00 pm, he is 455 miles from home. graph the line that represents justin's distance from hope at a given time. find and interpret the slope of the line.
alternate exterior angles
lie on opposite sides of the traversal outside the lines
same-side interior angles
lie on the same side of the traversal between lines
corresponding angles
lie on the same side of the traversal one the same sides of lines
midpoint formula
midpoint = x₂ + x₁ / 2, y₂ + y₁ / 2
m∠AOC = m∠AOB + m∠COB 99° = m∠AOB + 37° 62° = m∠AOB
m∠COB = 37° and m∠AOC = 99°. find m∠AOB.
x = midpoint of AB = (-2 + 1 / 2, 1 + 2 / 2) = (-½, ⁴/₂) = (-½, 2) y = midpoint of CB = (-2 + 4 / 2, 1 + 2 / 2) = (²/₂, ³/₂) = (1, 1½) CX is horizontal so its equation is y = 2 AY is vertical so its equation is x = 1 the centroid is (1, 2)
the diagram shows a new kind of triangular bread. where should the baker place her hand while spinning the dough so that the triangle is balanced?
247 + 114 > x 361 > x 114 + x > 247 x > 113 247 + x >114 x > -133 range is 361 > x > 113
the diagram shows the approximate distances from smithville to jackson and from rose city to jackson. what is the range of distances from smithville to rose city?
ST = UR ST = 37 XT = RX XT = 21 m∠RST + m∠STU = 180° m∠RST + 73.7° = 180° m∠RST = 106.3°
the diagram shows the parallelogram-shaped component that attaches a car's rearview mirror to the car. In parallelogram RSTU, UR = 37, RX = 21 and m∠STU = 73.7 degrees. Find ST, XT, and m∠RST
run
the difference in the x-values of two points on a line
rise
the difference in the y-values of two points on a line
circumference
the distance around a circle
sides
the incenter of a triangle is equidistant from the ______ of the triangle (incenter thrm.)
distance from a point to a line
the length of the perpendicular segment from the point to the line
remote interior angle
the measure of an exterior angle of a triangle is equal to the sum of the measures of its _________________ (exterior ∠ thrm.)
60°
the measure of each angle of an equilangular triangle is ___ (cor. 4-2-3)
parallel, 1/2
the midsegment of a trapezoid is ______________ to each base, and its length is _____ the sum of the lengths of the bases (trapezoid midsegment thrm.)
area
the number of nonoverlapping square units of a given size that exactly cover the figure
coordinate
the number that corresponds to a point on a number line
preimage
the original figure in a transformation
segment
the part of a line consisting of two points and all points between them
centroid of a triangle
the point of concurrency of the medians of a triangle
orthocenter of a triangle
the point of concurrency of the three altitudes of a triangle
incenter of a triangle
the point of concurrency of the three angle bisectors of a triangle
circumcenter of a triangle
the point of concurrency of the three perpendicular bisectors of a triangle
one-to-one
the points on a line can be put into a ___________ correspondence with the real numbers (ruler post.)
pi
the ratio of the circumference of a circle to its diameter
slope
the ratio of the rise to run
2(4) + 6(2) = 8 x 12 = 20in 20in x 20 = 400 in.
the rectangles on the panes of the windows of a building are 4 in. wide and 6 in. long. the perimeter of each rectangle is made by metal framing. if there are 20 rectangles in one window, how much metal framing will be needed?
