Download Solve using Geometric Relationships

Solve using Geometric Relationships
1. An exterior angle of a triangle measures 90°. If the remote interior angles are
congruent, then what is the measure of one of them?
A. 45°
B. 40°
C. 57.5°
D. 90°
2. Given: j
k and m
n.
*Note: Picture is not drawn to scale.
If m 1 = (4r + 98)° and m 2 = (-12r + 12)°, what is the value of r?
A. -8.75
B. 8.75
C. 5.375
D. -5.375
3.
In rectangle PQRS, if m
G = 38°, what is m
F?
A. 104°
B. 156°
C. 142°
D. 76°
4. Given: j
k and m
n.
*Note: Picture is not drawn to scale.
If m 1 = (24r + 90)° and m 2 = (12r)°, what is the value of r?
A. 7.5
B. -2.5
C. 2.5
D. -7.5
5.
If
T has a measure of 74°, what is the measure of
A. 32°
B. 106°
C. 148°
D. 74°
Q?
6.
Given the following measurements, what is the area of trapezoid FGJL?
A.
B.
C.
D.
7.
If m
A. 50°
B. 37.5°
C. 75°
D. 51°
f = 129° and m
*picture not drawn to scale
a = 54°, what is m c?
8.
Quadrilateral KLMN has the following angle measures.
What is the value of x?
A.
B.
C.
D.
9. The remote interior angles of a triangle measure 30° and 70°. What is the
measure of the exterior angle?
A. 80°
B. 40°
C. 100°
D. 130°
10.
Note: Figure not drawn to scale.
If m
A = x2, m
C = 133°, and m
D = 12x, what is m
A?
A. 133°
B. 131°
C. 49°
D. 47°
11.
*Note: Figure not drawn to scale.
In quadrilateral ABCD, m ABE = 40° and m AEB = 100°. What is the measure
of ECD?
A. 40°
B. 45°
C. 90°
D. 150°
12.
Note: Figure not drawn to scale.
If m K is twice m H, what is m H?
A. 20°
B. 30°
C. 60°
D. 45°
13.
Given the following measurements, what is the area of trapezoid ABCE?
A.
B.
C.
D.
14.
Note: Figure not drawn to scale.
If m
A = 44° and m
C = 119°, what is m
E?
A. 105°
B. 61°
C. 15°
D. 75°
15.
Note: Figure not drawn to scale.
If m R = 2x2 - 7°, m S = 4x2 - 6x + 40°, and m T = 18x + 3°, what is m R?
A. 11°
B. 43°
C. 65°
D. 25°
16. Two angles of a triangle have measures of 45° and 85°. Which of the following
could not be a measure of an exterior angle of the triangle?
A. 135°
B. 95°
C. 130°
D. 50°
17.
If m
f = 122°, m
*picture not drawn to scale
c = 3x, and m a = 2x, what is the measure of
A. 25°
B. 48.8°
C. 58°
D. 37.5°
18. Given: e
f and g
h.
a?
*Note: Picture is not drawn to scale.
If m 1 = (-6t + 26)° and m 2 = (-2t + 34)°, what is the value of t?
A. -15
B. 15
C. 2
D. -2
19.
*picture not drawn to scale
EFG
JKL
If m E = 6x, m K = 4x, and m L = 50°, what is the measure of angle J?
A. 72°
B. 84°
C. 52°
D. 78°
20.
*Note: Figure not drawn to scale.
In isosceles trapezoid ABCD, m BCD = 76° and m ABE = 52°. What is the
measure of ADE?
A. 14°
B. 24°
C. 166°
D. 52°
21.
Note: Figure not drawn to scale.
If m A = 55°, what is m D?
A. 145°
B. 35°
C. 125°
D. 55°
22.
Note: Figure not drawn to scale.
If m J = 56° and m K = 120°, what is m G?
A. 64°
B. 84°
C. 36°
D. 116°
23.
Note: Figure not drawn to scale.
The large triangle is an isosceles triangle. If m
Z = 142°, what is m
W?
A. 104°
B. 71°
C. 37°
D. 52°
24.
Note: Figure not drawn to scale.
If m F = 65°, m J = 24°, m K = 103°, and m L = 71°, what is m z?
A. 82°
B. 83°
C. 86°
D. 85°
25.
In parallelogram RSTU, if m U = 44°, what is m S?
A. 136°
B. 44°
C. 88°
D. 68°
26. Mr. Gray cuts pieces of yarn and puts them in groups of three. Without forming a
triangle, the students must decide if the yarn can make a triangle by just
measuring the lengths of the yarn. Which of the following combination of pieces
could be combined to make a triangle?
15 inches - 43 inches - 13 inches
5 inches - 29 inches - 7 inches
15 inches - 13 inches - 14 inches
5 inches - 34 inches - 18 inches
A. 5 inches - 34 inches - 18 inches
B. 5 inches - 29 inches - 7 inches
C. 15 inches - 43 inches - 13 inches
D. 15 inches - 13 inches - 14 inches
27.
*picture not drawn to scale
If m A = 118°, m D = 74° and m E = 30°, what is m C?
A. 42°
B. 36°
C. 58°
D. 132°
28. Given: w
x and y
z.
*Note: Picture is not drawn to scale.
If m 1 = (3s)° and m 2 = (6s - 30)°, what is the value of s?
A. 10
B. 3
C. -3
D. -10
29.
*Note: Figure not drawn to scale.
In rhombus ABCD, segments AC and BD are diagonals. If m DAB = 152°, what
is the measure of ABE?
A. 166°
B. 104°
C. 14°
D. 194°
30. Mrs. Jones' class is calculating the lengths of 3 sides of a triangle. If two of the
sides measure 5 and 28, which inequality represents the length of the third side?
A. x > 33
B. x > 23
C. 23 < x < 33
D. 23 < x < 33
31. Given: r
s and t
u.
*Note: Picture is not drawn to scale.
If m 1 = (6x + 34)° and m 2 = (4x + 26)°, what is the value of x?
A. -4
B. 12
C. 4
D. -12
32.
*picture not drawn to scale
If x is 6, and w =6
2, what is the value of y?
A. 36
B. 7
C. 49
D. 6
33.
In parallelogram RSTU, if m U = 42°, what is m T?
A. 42°
B. 138°
C. 84°
D. 69°
34.
In the figure above, FG
length of KJ?
KJ. If FH = 60, HJ = 45, and FG = 48, what is the
A. 38.4
B. 64
C. 60
D. 36
35.
*picture not drawn to scale
If x is 20 units, and y = 48 units, what is the value of w?
A. 48 units
B. 117 units
C. 60 units
D. 52 units
36.
Trapezoid ABCD has the following angle measures.
What is the value of x?
A.
B.
C.
D.
37.
In parallelogram LMNO, if m MNL = 25° and m NLM = 35°, then what is
m LNO?
A. 35°
B. 10°
C. 50°
D. 30°
38.
Note: Figure not drawn to scale.
If m T = 75° and m Q = 160°, what is m S?
A. 90°
B. 80°
C. 95°
D. 85°
39.
Note: Figure not drawn to scale.
If
ZXY is an equilateral triangle and m
W = 22°, what is m
WXZ?
A. 31°
B. 98°
C. 58°
D. 38°
40. What is m 1 if
A. 93°
B. 38°
C. 55°
D. 35°
= 38° and β = 87°?
Answers
1. A
2. A
3. A
4. C
5. C
6. B
7. C
8. B
9. C
10. C
11. A
12. C
13. A
14. D
15. D
16. D
17. B
18. A
19. D
20. B
21. D
22. A
23. D
24. B
25. B
26. D
27. A
28. A
29. C
30. C
31. B
32. D
33. B
34. D
35. D
36. C
37. A
38. D
39. D
40. D
Explanations
1.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to
the sum of the measures of the remote interior angles.
The angles are congruent so divide 90° by 2 to find the solution.
The answer is 45°.
2.
Angle 1 and angle 2 are supplementary angles.
So, set the sum of the two expressions equal to 180°, and solve for r.
m 1 + m 2 = 180°
(4r + 98)° + (-12r + 12)° = 180°
4r + 98 - 12r + 12 = 180
-8r + 110 = 180
-8r = 70
r = -8.75
Therefore, the value of r is -8.75.
3.
Since PQRS is a rectangle, the diagonals are congruent.
Since a rectangle is a parallelogram, the diagonals bisect each other.
Therefore, SF
E and G.
RF and
SFR is an isosceles triangle with congruent base angles
So,
m
4.
E+m
G+m
F = 180°
38° + 38° + m
F = 180°
76° + m
F = 180°
m
F = 180° - 76°
m
F = 104°
Angle 1 and angle 2 are supplementary angles.
So, set the sum of the two expressions equal to 180°, and solve for r.
m 1 + m 2 = 180°
(24r + 90)° + (12r)° = 180°
24r + 90 + 12r = 180
36r + 90 = 180
36r = 90
r = 2.5
Therefore, the value of r is 2.5.
5.
Notice that RST is an isosceles triangle. Since
S also has a measure of 74°.
T has a measure of 74°,
Since the interior angles of a triangle must equal 180°, the measure of
R = 180° - 74° - 74° = 32°.
Notice that Q is the supplement of
32° = 148°.
6.
R. Therefore,
Q = 180° -
R = 180° -
The formula for the area of a trapezoid is shown below.
Let side FG be base1. The length of side FG is given.
Let side JL be base2. Calculate the length of side JL.
The height of the trapezoid is given.
Find the area of the trapezoid.
7.
Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal
to the sum of the measures of the remote interior angles.
This indicates that m
f=m
a+m
c. Therefore,
129° = 54° + m
c
75° = m c
8.
The sum of the interior angles of a quadrilateral is 360°.
Add up the degree measures of all four angles, and solve for x.
9.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal
to the sum of the measures of the remote interior angles.
The answer is 100°.
10.
The triangle sum theorem states that the sum of the measures of the angles of a
triangle is 180°.
m A + m B + m D = 180°
The m B is unknown. Since m C is known, m B can be found using the linear
pairs theorem.
m
m
B+m
C = 180°
B + 133° = 180°
m
B = 47°
Since the sum of the interior angles of a triangle equals 180°, solve the following
for x.
m
A+m
B+m
D = 180°
x + 47° + 12x = 180°
2
x2 + 12x - 133° = 0°
(x - 7°)(x + 19°) = 0°
x = 7°, (-19)°
Since all of the angles are positive angles, x = 7°.
The m
m A.
A is found by substituting the value of x into the equation representing
m
A = x2
= (7°)2
= 49°
11.
The Alternate Interior Angle Theorem states that if two parallel lines are cut by a
transversal, then the alternate interior angles will be congruent.
Since m ABC is 90° and m BCD is 90°, it can be assumed that AB is parallel to
DC. It is seen that the two parallel segments are cut by the transversal BD;
therefore, m ABE = m CDE.
Since m ABE = m CDE and m ABE = 40°, m CDE = 40°.
Since
AEB and
DEC are vertical angles, their measures will be congruent.
Since m AEB = 100°, m DEC = 100°.
To find the measure of
ECD, use triangle DEC.
m CDE + m DEC + m ECD = 180°
40° + 100° + m ECD = 180°
140° + m ECD = 180°
m ECD = 40°
The measure of
12.
Since
H and
ECD equals 40°.
K are supplementary angles, m H + m K = 180°.
Since m K is twice m H, set up an equation and solve for m H.
m H + m K = 180°
m H + 2(m H) = 180°
3(m H) = 180°
m H = 60°
13.
The formula for the area of a trapezoid is shown below.
Let side AB be base1. The length of side AB is given.
Let side EC be base2. Calculate the length of side EC.
The height of the trapezoid is given.
Find the area of the trapezoid.
14.
To calculate m E, the interior angles of the triangle must first be found.
Since
B and
C are supplementary angles, m B + m C = 180°. Find m B.
m B + m C = 180°
m B + 119° = 180°
m B = 61°
Find m D. Remember that the sum of the measures of the angles of a triangle is
180°.
m A + m B + m D = 180°
44° + 61° + m D = 180°
m D = 75°
Find m E. Since
D and
E are vertical angles, m D = m E.
m D=m E
75° = m E
15.
Since the sum of the interior angles of a triangle equals 180°, solve the following
for x.
m R + m S + m T = 180°
2x2 - 7° + 4x2 - 6x + 40° + 18x + 3° = 180°
6x2 + 12x + 36° = 180°
x2 + 2x + 6° = 30°
x2 + 2x - 24° = 0°
(x + 6°)(x - 4°) = 0°
x = (-6)°, 4°
Since all of the angles are positive angles, x = 4°.
The m R is found by substituting x = 4° into the equation representing m R.
2(x)2 - 7° = 2(4°)2 - 7°
= 2(16°) - 7°
= 32° - 7°
= 25°
16.
The triangle sum theorem states that the sum of the measures of the angles of a
triangle is 180°.
So, the measure of the third angle is 180° - 45° - 85° = 50°.
The exterior angle theorem states that the measure of an exterior angle of a
triangle is equal to the sum of the measures of the remote interior angles.
exterior angle 1 = 85° + 50° = 135°
exterior angle 2 = 45° + 50° = 95°
exterior angle 3 = 85° + 45° = 130°
Therefore, 50° is not an exterior angle of the triangle.
17.
Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal
to the sum of the measures of the remote interior angles.
This indicates that m
f=m
c+m
a. Therefore,
122° = 3x + 2x
122° = 5x
24.4° = x
a = 2x
a = 2(24.4)
a = 48.8°
18.
Angle 1 and angle 2 are supplementary angles.
So, set the sum of the two expressions equal to 180°, and solve for t.
m 1 + m 2 = 180°
(-6t + 26)° + (-2t + 34)° = 180°
-6t + 26 - 2t + 34 = 180
-8t + 60 = 180
-8t = 120
t = -15
Therefore, the value of t is -15.
19.
E and J,
to solve.
F and
K, and
G and
J+
L are congruent, so use
K+
J and
K and
L
L = 180°
6x + 4x + 50 = 180°
10x = 130°
x = 13°
J = 6x
J = 6(13°)
J = 78°
20.
Given that trapezoid ABCD is isosceles, the base angles are congruent; therefore,
m ADC = m BCD.
Since m ADC = m BCD and m BCD = 76°, m ADC = 76°.
The Alternate Interior Angle Theorem states that if two parallel lines are cut by a
transversal, then the alternate interior angles will be congruent.
Since figure ABCD is a trapezoid, AB is parallel to DC. It is seen that the two
parallel segments are cut by the transversal BD; therefore, m ABE = m CDE.
Since m ABE = m CDE and m ABE = 52°, m CDE = 52°.
Given that m ADC = m ADE + m CDE, find m ADE.
m ADC = m ADE + m CDE
76° = m ADE + 52°
24° = m ADE
The measure of
21.
ADE equals 24°.
To calculate m D, m C must be known.
There is currently not enough information to find m C. Although, there is enough
information to solve for m B; therefore, find the m B in the triangle on the left.
Remember that the sum of the measures of the angles of a triangle is 180°.
m A + m B + 90° = 180°
55° + m B + 90° = 180°
m B = 35°
The altitude to the base of an isosceles triangle is also an angle bisector;
therefore, m B = m C in the given triangle. The following is true for C.
m B=m C
35° = m C
Since
C and
D are complementary, m C + m D = 90°.
m C + m D = 90°
35° + m D = 90°
m D = 55°
22.
If m J = 56°, then m F = 56° by the definition of vertical angles.
The exterior angle theorem states that the measure of an exterior angle of a
triangle is equal to the sum of the measures of the remote interior angles.
m K=m G+m F
120° = m G + 56°
64° = m G
23.
Since
Z and
Y form a straight line, m
So, m
Y = 180° - m
Y+m
Z = 180°.
Z = 180° - 142° = 38°.
Since the larger triangle is an isosceles triangle,
m Y = m V = 38° and m
Now, set up an equation and solve for m W.
m
V+m
38° + m
W+m
W+m
X+m
W=m
X.
Y = 180°
W + 38° = 180°
76° + 2(m
W) = 180°
2(m
W) = 104°
m
W = 52°
24.
In the given figure, x is the measure of an angle of GJK. Since the sum of the
measures of the angles of a triangle is 180°, set up an equation and solve for
m x.
m x + m J + m K = 180°
m x + 24° + 103° = 180°
m x = 180° - 127°
m x = 53°
In the given figure, y is the measure of an angle of FHL. Since the sum of the
measures of the angles of a triangle is 180°, set up an equation and solve for
m y.
m y + m F + m L = 180°
m y + 65° + 71° = 180°
m y = 180° - 136°
m y = 44°
Since m x = 53° and m y = 44°, set up an equation and solve for m z.
m x + m y + m z = 180°
53° + 44° + m z = 180°
m z = 180° - 97°
m z = 83°
25.
In a parallelogram, opposite angles are congruent.
Therefore,
m U=m S
44° = m S
26.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a
triangle is greater than the length of the third side.
The only group which satisfies the theorem is
15 inches - 13 inches - 14 inches.
15 inches + 13 inches > 14 inches
15 inches + 14 inches > 13 inches
14 inches + 13 inches > 15 inches
27.
Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal
to the sum of the measures of the remote interior angles.
To solve, find the value of F using the angle measure of a straight line and use
the Exterior Angle Theorem to find C.
D+
E+
F = 180° because they make a straight line.
F = 180° - 74° - 30°
F = 76°
C+
F=
A
C=
A-
F
C = 118° - 76°
C = 42°
28.
Angle 1 and angle 2 are congruent angles.
So, set the two expressions equal to each other, and solve for s.
m 1=m 2
(3s)° = (6s - 30)°
3s = 6s - 30
30 = 6s - 3s
30 = 3s
10 = s
Therefore, the value of s is 10.
29.
Given that figure ABCD is a rhombus, diagonal AC and diagonal BD must bisect
opposite angles.
Therefore, m DAE = m BAE.
Given that m DAE = m BAE and m DAB = 152°, the following is true.
m DAE + m BAE = m DAB
m DAE + m BAE = 152°
m BAE + m BAE = 152°
2 × m BAE = 152°
m BAE = 76°
Since segment AC and segment BD are diagonals, it is also true that segment AC
is perpendicular to segment BD; therefore, m AEB = 90°.
To find the measure of
ABE, use the triangle ABE.
m BAE + m AEB + m ABE = 180°
76° + 90° + m ABE = 180°
166° + m ABE = 180°
m ABE = 14°
The measure of
30.
ABE equals 14°.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a
triangle is greater than the length of the third side.
Therefore, the sum of the two given sides must be greater than the third side, x.
5 + 28 > x
33 > x
x + 5 > 28
x > 23
The answer is 23 < x < 33
31.
Angle 1 and angle 2 are supplementary angles.
So, set the sum of the two expressions equal to 180°, and solve for x.
m 1 + m 2 = 180°
(6x + 34)° + (4x + 26)° = 180°
6x + 34 + 4x + 26 = 180
10x + 60 = 180
10x = 120
x = 12
Therefore, the value of x is 12.
32.
Use the Pythagorean Theorem to solve for y.
w2 = x2 + y2
2
2
22 = 6 + y
6
72 = 36 + y2
36 = y2
36 =
y2
6=y
33.
In a parallelogram, consecutive angles are supplementary.
Therefore,
m U + m T = 180°
42° + m T = 180°
m T = 180° - 42°
m T = 138°
34.
Notice that
GHF and
KHJ are vertical angles. Therefore,
Since FG
KJ, HFG and
HFG
HJK.
GHF
KHJ.
HJK are alternate interior angles. Therefore,
Given two triangles, if there are two pairs of congruent corresponding angles, then
the remaining corresponding angles must also be congruent. So, all three angles
of one triangle are congruent to the three angles of the other triangle. Thus,
FGH is similar to JKH.To find the length of KJ solve the following proportion.
FH
HJ
60
45
=
=
FG
KJ
48
KJ
60KJ
=
2,160
KJ
=
36
35.
Use the Pythagorean Theorem to solve for w.
w2 = x2 + y2
w2 = 202 + 482
w2 = 400 + 2,304
w2 = 2,704
w2 =
2,704
w = 52 units
36.
The sum of the interior angles of a quadrilateral is 360°.
Add up the degree measures of all four angles, and solve for x.
37.
The Alternate Interior Angle Theorem states that if two parallel lines are cut by a
transversal, then the alternate interior angles will be congruent.
Since m NLM = 35°, m LNO = 35°.
38.
The exterior angle theorem states that the measure of an exterior angle of a
triangle is equal to the sum of the measures of the remote interior angles.
This indicates that m Q = m T + m S. Use this equation to find m S.
m Q=m T+m S
160° = 75° + m S
85° = m S
39.
Since
ZXY is an equilateral triangle, m
Therefore, m
XZY = 60°.
XZW = 180° - 60° = 120°.
Since the sum of the interior angles of a triangle is 180°, set up an equation and
solve for m WXZ.
m
WXZ = 180° - m
W-m
= 180° - 22° - 120°
= 38°
XZW
40.
The sum of the interior angles of a triangle is 180°.
So, 180° - (38° + 87°) = 55°.
To find m 1, use the same theorem: 180° - (90° + 55°) = 35°.
Therefore, m 1 = 35°.