Download ExamView - Milestone Review unit 1.tst

Name: ________________________ Class: ___________________ Date: __________
ID: A
GA Milestone Review Unit 1
____
1. The map shows a linear section of Highway 35. Today, the Ybarras plan to drive the 360 miles from
Springfield to Junction City. They will stop for lunch in Roseburg, which is at the midpoint of the trip. If they
have already traveled 55 miles this morning, how much farther must they travel before they stop for lunch?
a.
b.
125 mi
145 mi
c.
d.
180 mi
305 mi
____
2. K is the midpoint of JL . JK = 6x and KL = 3x + 3 . Find J K, KL, and J L.
a. J K = 1, KL = 1, J L = 2
c. J K = 12, KL = 12, J L = 6
b. J K = 6, KL = 6, J L = 12
d. J K = 18, KL = 18, J L = 36
____
3. BD bisects ∠ABC , m∠ABD = (7x − 1)°, and m∠DBC = (4x + 8)°. Find m∠ABD.
a. m∠ABD = 22°
c. m∠ABD = 40°
b. m∠ABD = 3°
d. m∠ABD = 20°
____
4. Two angles with measures (2x 2 + 3x − 5)° and (x 2 + 11x − 7)° are supplementary. Find the value of x and the
measure of each angle.
a. x = 5; 60°; 30°
c. x = 5; 60°; 120°
b. x = 6; 85°; 95°
d. x = 4; 40°; 90°
____
5. Two lines intersect to form two pairs of vertical angles. ∠1 with measure (20x + 7)° and ∠3 with measure
(5x + 7y + 49)° are vertical angles. ∠2 with measure (3x − 2y + 30)° and ∠4 are vertical angles. Find the
values x and y and the measures of all four angles.
a. x = 6; y = 10; 127°; 127°; 28°; 28°
c. x = 5; y = 5; 107°; 107°; 73°; 73°
b. x = 8; y = 11, 167°; 167°; 13°; 13°
d. x = 7; y = 9; 147°; 147°; 33°; 33°
→
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Name: ________________________
____
6. Write a justification for each step, given that EG = FH .
EG = FH
EG = EF + FG
FH = FG + GH
EF + FG = FG + GH
EF = GH
a.
b.
c.
d.
____
ID: A
Given information
[1]
Segment Addition Postulate
[2]
Subtraction Property of Equality
[1] Angle Addition Postulate
[2] Subtraction Property of Equality
[1] Substitution Property of Equality
[2] Transitive Property of Equality
[1] Segment Addition Postulate
[2] Definition of congruent segments
[1] Segment Addition Postulate
[2] Substitution Property of Equality
7. Find m∠ABC .
a.
b.
m∠ABC = 40°
m∠ABC = 45°
c.
d.
2
m∠ABC = 35°
m∠ABC = 50°
Name: ________________________
____
8. Write and solve an inequality for x.
a.
b.
____
ID: A
c.
d.
x>2
x<2
x>1
x < −2
9. Write a justification for each step.
m∠JKL = 100°
m∠JKL = m∠JKM + m∠MKL
100° = (6x + 8)° + (2x − 4)°
100 = 8x + 4
96 = 8x
12 = x
x = 12
a.
b.
c.
d.
[1]
Substitution Property of Equality
Simplify.
Subtraction Property of Equality
[2]
Symmetric Property of Equality
[1] Transitive Property of Equality
[2] Division Property of Equality
[1] Angle Addition Postulate
[2] Division Property of Equality
[1] Angle Addition Postulate
[2] Simplify.
[1] Segment Addition Postulate
[2] Multiplication Property of Equality
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Name: ________________________
ID: A
____ 10. A gardener has 26 feet of fencing for a garden. To find the width of the rectangular garden, the gardener uses
the formula P = 2l + 2w , where P is the perimeter, l is the length, and w is the width of the rectangle. The
gardener wants to fence a garden that is 8 feet long. How wide is the garden? Solve the equation for w, and
justify each step.
P = 2l + 2w
26 = 2(8) + 2w
26 = 16 + 2w
−16 = −16
10 = 2w
10
2w
=
2
2
5=w
w=5
a.
b.
Given equation
[1]
Simplify.
Subtraction Property of Equality
Simplify.
[2]
Simplify.
Symmetric Property of Equality
[1] Substitution Property of Equality
[2] Division Property of Equality
The garden is 5 ft wide.
[1] Simplify
[2] Division Property of Equality
The garden is 5 ft wide.
c.
d.
[1] Substitution Property of Equality
[2] Subtraction Property of Equality
The garden is 5 ft wide.
[1] Subtraction Property of Equality
[2] Simplify
The garden is 5 ft wide.
____ 11. Find m∠RST .
a.
b.
m∠RST = 108°
m∠RST = 24°
c.
d.
m∠RST = 156°
m∠RST = 72°
____ 12. Three vertices of parallelogram WXYZ are X(–2,–3), Y(0, 5), and Z(7, 7). Find the coordinates of vertex W.
a. (4, 0)
c. (5, 0)
b. (9, 15)
d. (5, –1)
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Name: ________________________
ID: A
____ 13. Fill in the blanks to complete the two-column proof.
Given: ∠1 and ∠2 are supplementary. m∠1 = 135°
Prove: m∠2 = 45°
Proof:
Statements
1. ∠1 and ∠2 are supplementary.
2. [1]
3. m∠1 + m∠2 = 180°
4. 135° + m∠2 = 180°
5. m∠2 = 45°
a.
b.
c.
d.
Reasons
1. Given
2. Given
3. [2]
4. Substitution Property
5. [3]
[1] m∠2 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
[1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Substitution Property
[1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
[1] m∠1 = 135°
[2] Definition of complementary angles
[3] Subtraction Property of Equality
____ 14. A video game designer is modeling a tower that is 320 ft high and 260 ft wide. She creates a model so that
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the similarity ratio of the model to the tower is 500
. What is the height and the width of the model in inches?
a.
b.
c.
d.
height = 0.64 in.; width = 0.52 in.
height = 3840 in.; width = 3120 in.
height = 7.68 in.; width = 6.24 in.
height = 160,000 in.; width = 130,000 in.
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Name: ________________________
ID: A
____ 15. Use the given flowchart proof to write a two-column proof of the statement AF ≅ FD .
Flowchart proof:
AB = CD ;
BF = FC
Given
AB + BF = AF
FC + CD = FD
Segment
Addition
Postulate
AB + BF =
FC + CD
Addition
Property of
Equality
AF = FD
AF ≅ FD
Substitution
Definition of
congruent segments
Complete the proof.
Two-column proof:
Statements
1. AB = CD ; BF = FC
2. [1]
3. [2]
4. AF = FD
5. AF ≅ FD
a.
b.
c.
d.
[1]
[2]
[1]
[2]
[1]
[2]
[1]
[2]
Reasons
1. Given
2. Addition Property of Equality
3. Segment Addition Postulate
4. Substitution
5. Definition of congruent segments
AB + BF = AF ; FC + CD = FD
AF = FD
AF = FD
AB + BF = FC + CD
AB = CD ; BF = FC
AB + BF = FC + CD
AB + BF = FC + CD
AB + BF = AF ;FC + CD = FD
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Name: ________________________
ID: A
____ 16. Use the given two-column proof to write a flowchart proof.
Given: ∠1 ≅ ∠4
Prove: m∠2 = m ∠3
Two-column proof:
Statements
1. ∠1 ≅ ∠4
2. ∠1 and ∠2 are supplementary. ∠3 and ∠4
are supplementary.
3. ∠2 ≅ ∠3
4. m∠2 = m ∠3
Reasons
1. Given
2. Definition of linear pair
3. Congruent Supplements Theorem
4. Definition of congruent segments
Complete the proof.
Flowchart proof:
∠1 ≅ ∠4
Given
[1]
∠2 ≅ ∠3
Definition of linear pair
a.
b.
c.
d.
[2]
[1] ∠1 and ∠2 are supplements; ∠3 and ∠4 are supplementary
[2] Congruent Complements Theorem
[1] ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary
[2] Congruent Supplements Theorem
[1] ∠2 ≅ ∠3
[2] Definition of congruent segments
[1] Definition of congruent segments
[2] Congruent Supplements Theorem
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m∠2 = m ∠3
Definition of
congruent segments
Name: ________________________
ID: A
____ 17. Use the given paragraph proof to write a two-column proof.
Given: ∠BAC is a right angle. ∠1 ≅ ∠3
Prove: ∠2 and ∠3 are complementary.
Paragraph proof:
Since ∠BAC is a right angle, m∠BAC = 90° by the definition of a right angle. By the Angle Addition
Postulate, m∠BAC = m∠1 + m∠2 . By substitution, m∠1 + m∠2 = 90°. Since ∠1 ≅ ∠3, m∠1 = m∠3 by the
definition of congruent angles. Using substitution, m∠3 + m∠2 = 90°. Thus, by the definition of
complementary angles, ∠2 and ∠3 are complementary.
Complete the proof.
Two-column proof:
Statements
1. ∠BAC is a right angle. ∠1 ≅ ∠3
2. m∠BAC = 90°
3. m∠BAC = m∠1 + m∠2
4. m∠1 + m∠2 = 90°
5. m∠1 = m∠3
6. m∠3 + m∠2 = 90°
7. ∠2 and ∠3 are complementary.
Reasons
1. Given
2. Definition of a right angle
3. [1]
4. Substitution
5. [2]
6. Substitution
7. Definition of complementary angles
a.
c.
b.
[1] Substitution
[2] Definition of congruent angles
[1] Angle Addition Postulate
[2] Definition of congruent angles
d.
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[1] Angle Addition Postulate
[2] Definition of equality
[1] Substitution
[2] Definition of equality
Name: ________________________
ID: A
____ 18. Find the value of x.
a.
b.
x=6
x=4
____ 19. Determine whether triangles
a.
b.
c.
d.
c.
d.
EFG and
The triangles are congruent because
(x, y) → (−x, y).
The triangles are congruent because
(x, y) → (−y, −x).
The triangles are congruent because
(x, y) → (x, −y).
The triangles are congruent because
(x, y) → (−y, x).
x=2
x=8
PQR are congruent.
EFG can be mapped to
PQR by a reflection:
EFG can be mapped to
PQR by a rotation:
EFG can be mapped to
PQR by a reflection:
EFG can be mapped to
PQR by a rotation:
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Name: ________________________
ID: A
____ 20. Given that ∆ABC ≅ ∆DEC and m∠E = 23°, find m∠ACB.
a.
b.
m∠ACB = 77°
m∠ACB = 67°
c.
d.
m∠ACB = 23°
m∠ACB = 113°
c.
d.
m∠K = 79°
m∠K = 39°
____ 21. Find m∠K .
a.
b.
m∠K = 63°
m∠K = 55°
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Name: ________________________
ID: A
____ 22. Apply the transformation M to the triangle with the given vertices.
Identify and describe the transformation.
M: (x, y) → (x – 6, y + 2)
E(3, 0), F(1, –2), G(5, –4)
a.
c.
This is a translation 6 units left and 2
units up.
b.
This is a translation 6 units left.
d.
This is a translation 2 units left and 6
units up.
This is a translation 6 units left and 2
units down.
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Name: ________________________
ID: A
____ 23. Apply the transformation M to the polygon with the given vertices.
Identify and describe the transformation.
M: (x, y) → (–x, –y)
A(–3, 6), B(–3, 1), C(1, 1), D(1, 6)
a.
c.
This is a reflection over the x-axis.
This is a rotation of 180° about the
origin.
b.
d.
This is a rotation of 90° clockwise about
the origin.
This is a rotation of 180° about the
origin.
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Name: ________________________
ID: A
____ 24. Find m∠E and m∠N , given m∠F = m∠P, m∠E = (x 2 )°, and m∠N = (4x 2 − 75)°.
a.
b.
m∠E = 25°, m∠N = 25°
m∠E = 25°, m∠N = 65°
c.
d.
m∠E = 65°, m∠N = 25°
m∠E = 65°, m∠N = 65°
____ 25. Given: P is the midpoint of TQ and RS .
Prove: ∆TPR ≅ ∆QPS
Complete the proof.
Proof:
Statements
1. P is the midpoint of TQ and RS .
Reasons
1. Given
2. TP ≅ QP, RP ≅ SP
3. [2]
4. ∆TPR ≅ ∆QPS
2. [1]
a.
c.
b.
[1]. Definition of midpoint
[2] ∠TPR ≅ ∠QPS
[3] SAS
[1] Definition of midpoint
[2] RT ≅ SQ
[3] SSS
3. Vertical Angles Theorem
4. [3]
d.
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[1] Definition of midpoint
[2] ∠PRT ≅ ∠PSQ
[3] SAS
[1] Definition of midpoint
[2] ∠TPR ≅ ∠QPS
[3] SSS
Name: ________________________
ID: A
____ 26. What additional information do you need to prove ∆ABC ≅ ∆ADC by the SAS Postulate?
a.
b.
AB ≅ AD
∠ACB ≅ ∠ACD
c.
d.
∠ABC ≅ ∠ADC
BC ≅ DC
____ 27. Determine if you can use ASA to prove ∆CBA ≅ ∆CED. Explain.
a.
b.
c.
d.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. No other congruence
relationships can be determined, so ASA cannot be applied.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Adjacent
Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles
Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles
Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by SAS.
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Name: ________________________
ID: A
____ 28. For these triangles, select the triangle congruence statement and the postulate or theorem that supports it.
a.
b.
∆ABC ≅ ∆JLK , HL
∆ABC ≅ ∆JKL, HL
c.
d.
∆ABC ≅ ∆JLK , SAS
∆ABC ≅ ∆JKL, SAS
____ 29. Two Seyfert galaxies, BW Tauri and M77, represented by points A and B, are equidistant from Earth,
represented by point C. What is m∠A?
a.
b.
m∠A = 65°
m∠A = 115°
c.
d.
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m∠A = 50°
m∠A = 60°
Name: ________________________
ID: A
____ 30. Given ∆ABC with AB = 3 , BC = 5 , and CA = 6, find the length of midsegment XY .
a.
b.
XY = 3
XY = 1.5
c.
d.
XY = 2.5
XY = 2
____ 31. Point O is the centroid of ∆ABC , BY = 3.3 and CO = 3 . Find BO .
a.
b.
BO = 2.2
BO = 1.1
c.
d.
BO = 3.3
BO = 3
→
____ 32. Given that YW bisects ∠XYZ and WZ = 4.23, find WX .
a.
b.
WX = 4.23
WX = 8.46
c.
d.
16
WX = 45°
WX = 90°
Name: ________________________
ID: A
____ 33. 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?
a.
ÁÊ 1, 1 ˜ˆ
Ë
¯
c.
b.
ÊÁ 1, 0 ˆ˜
Ë
¯
d.
ÁÁÁÊ
ÁË
ÁÁÁÊ
ÁË
1
2
3
2
ˆ
, 1 ˜˜˜˜
¯
ˆ˜˜
, 1 ˜˜
¯
____ 34. The diagram shows the parallelogram-shaped component that attaches a car’s rearview mirror to the car. In
parallelogram RSTU, UR = 25, RX = 16, and m∠STU = 42.4o. Find ST, XT, and m∠RST.
a.
b.
ST = 16, m∠RST = 42.4°, XT = 25
ST = 25, m∠RST = 47.8°, XT = 16
c.
d.
ST = 25, m∠RST = 137.6°, XT = 16
ST = 5, m∠RST = 137.6°, XT = 4
c.
d.
MP = 20
MP = 6
____ 35. MNOP is a parallelogram. Find MP.
a.
b.
MP = 25
MP = 30
17
Name: ________________________
ID: A
____ 36. An artist designs a rectangular quilt piece with different types of ribbon that go from the corner to the center
of the quilt. The dimensions of the rectangle are AB = 10 inches and AC = 14 inches. Find BX .
a.
b.
BX = 7 inches
BX = 10 inches
c.
d.
BX = 5 inches
BX = 14 inches
c.
d.
SU = 5
SU = 3
____ 37. TRSU is a rhombus. Find SU .
a.
b.
SU = 7
SU = 1
____ 38. Apply the dilation D to the polygon with the given vertices. Name the coordinates of the image points.
D: (x, y) → (3x, 3y)
J(1, 4), K(6, 4), L(6, 1), M(1, 1)
a.
b.
J´(12, 3), K´(12, 18),
L´(3, 18), M´(3, 3)
J´(–3, –12), K´(–18, –12),
L´(–18, –3), M´(–3, –3)
c.
d.
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J´(3, 12), K´(18, 12),
L´(18, 3), M´(3, 3)
J´(3, 12), K´(18, 12),
L´(6, 1), M´(1, 1)
Name: ________________________
ID: A
____ 39. To find out how wide a river is, Jon and Sally mark an X at the spot directly across from a big rock on the
other side of the river. Then they walk in a straight line along the river, perpendicular to the straight line
between the X and the rock. After walking for 20 feet Jon stops while Sally continues along the straight line
for another 10 feet. Then she makes a 90 degree turn and walks for 30 feet. When she stops and looks at the
rock she sees that the straight line from her to the rock passes through Jon. What is the distance from X to the
rock?
a.
b.
30 feet
50 feet
c.
d.
60 feet
63 feet
c.
d.
NP = 1.6
NP = 2
____ 40. Find NP.
a.
b.
NP = 1
NP = 1.25
19
Name: ________________________
ID: A
____ 41. An artist used perspective to draw guidelines in her picture of a row of parallel buildings. How many
centimeters is it from Point B to Point C?
a.
b.
1 cm
3.75 cm
c.
d.
4 cm
2.4 cm
c.
d.
BD = 10
BD = 12
____ 42. Find BD .
a.
b.
BD = 5
BD = 22
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Name: ________________________
ID: A
____ 43. Given that ∆KON ∼ ∆LOM, find the coordinates of L and the scale factor.
4
3
a.
L (6, 0) and scale factor is 2
c.
L (9, 0) and scale factor is
b.
L (9, 0) and scale factor is 3
d.
L (6, 0) and scale factor is 3
____ 44. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a.
b.
m∠1 = 95°
m∠1 = 80°
c.
d.
45. Find the value of x in the rhombus.
21
m∠1 = 85°
m∠1 = 75°
Name: ________________________
ID: A
46. Find the value of x so that m Ä n.
47. Find the value of n in the triangle.
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ID: A
GA Milestone Review Unit 1
Answer Section
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