Download ExamView - SCA 1 Review.tst

Name: ________________________ Class: ___________________ Date: __________
ID: A
Geometry Short Cycle 1 Exam Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
←
→
____
1. Name a plane that contains AC .
a.
b.
____
c.
d.
plane WRT
plane RCA
c.
d.
BC = 7
BC = 8
2. Find the length of BC .
a.
b.
____
plane ACR
plane WCT
BC = –7
BC = –9
3. The tip of a pendulum at rest sits at point B. During an experiment, a physics student sets the pendulum in
motion. The tip of the pendulum swings back and forth along part of a circular path from point A to point C.
During each swing the tip passes through point B. Name all the angles in the diagram.
a.
b.
∠AOB, ∠BOC
∠AOB, ∠COB, ∠AOC
c.
d.
1
∠AOB, ∠BOA, ∠COB, ∠BOC
∠OAB, ∠OBC , ∠OCB
Name: ________________________
____
4. Name all pairs of vertical angles.
a.
b.
c.
d.
____
∠MLN and ∠JLM ; ∠JLK and ∠KLN
∠JLK and ∠MLN ; ∠JLM and ∠KLN
∠JKL and ∠MNL; ∠JML and ∠KNL
∠JLK and ∠JLM ; ∠KLN and ∠MLN
5. There are four fruit trees in the corners of a square backyard with 30-ft sides. What is the distance between
the apple tree A and the plum tree P to the nearest tenth?
a.
b.
____
ID: A
42.4 ft
42.3 ft
c.
d.
30.0 ft
30.3 ft
6. Identify the transformation. Then use arrow notation to describe the transformation.
a.
b.
c.
d.
The transformation is a 90° rotation. ABC → A' B' C'
The transformation is a 45° rotation. ABC → A' B' C'
The transformation is a reflection. ABC → A' B' C'
The transformation is a translation. ABC → A' B' C'
2
Name: ________________________
____
ID: A
7. Name three collinear points.
a.
b.
P, G, and N
R, P, and N
c.
d.
R, P, and G
R, G, and N
____
8. Determine if the biconditional is true. If false, give a counterexample.
A figure is a square if and only if it is a rectangle.
a. The biconditional is true.
b. The biconditional is false. A rectangle does not necessarily have four congruent sides.
c. The biconditional is false. All squares are parallelograms with four 90° angles.
d. The biconditional is false. A rectangle does not necessarily have four 90° angles.
____
9. Solve the equation 4x − 6 = 34. Write a justification for each step.
4x − 6 = 34
Given equation
[1]
+6 +6
4x
= 40
Simplify.
4x
40
[2]
=
4
4
x = 10
Simplify.
a.
b.
[1] Substitution Property of Equality;
[2] Division Property of Equality
[1] Addition Property of Equality;
[2] Division Property of Equality
c.
d.
1
[1] Division Property of Equality;
[2] Subtraction Property of Equality
[1] Addition Property of Equality;
[2] Reflexive Property of Equality
Name: ________________________
ID: A
____ 10. 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
____ 11. Identify the property that justifies the statement.
AB ≅ CD and CD ≅ EF . So AB ≅ EF .
a. Reflexive Property of Congruence
c.
b. Substitution Property of Equality
d.
Symmetric Property of Congruence
Transitive Property of Congruence
____ 12. 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°
4
Name: ________________________
ID: A
____ 13. 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.
5
[1] Angle Addition Postulate
[2] Definition of equality
[1] Substitution
[2] Definition of equality
Name: ________________________
ID: A
____ 14. Identify the transversal and classify the angle pair ∠11 and ∠7.
a.
b.
c.
d.
The transversal is line l. The angles are corresponding angles.
The transversal is line l. The angles are alternate interior angles.
The transversal is line n. The angles are alternate exterior angles.
The transversal is line m. The angles are corresponding angles.
____ 15. Find m∠ABC .
a.
b.
m∠ABC = 40°
m∠ABC = 45°
c.
d.
6
m∠ABC = 35°
m∠ABC = 50°
Name: ________________________
ID: A
____ 16. 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 = 10, show that the lanes are parallel.
a.
b.
c.
d.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles and they are congruent, so the lanes are parallel
by the Alternate Interior Angles Theorem.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles, and they are congruent, so the lanes are parallel
by the Converse of the Alternate Interior Angles Theorem.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are corresponding angles and they are congruent, so the lanes are parallel by
the Converse of the Corresponding Angles Postulate.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are same-side interior angles and they are supplementary, so the lanes are
parallel by the Converse of the Same-Side Interior Angles Theorem.
____ 17. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
←

→
←
→
AB and CD for A(3,5), B(−2,7), C(10,5), and D(6,15)
a. neither
c. parallel
b. perpendicular
____ 18. Determine whether the lines 12x + 3y = 3 and y = 4x + 1 are parallel, intersect, or coincide.
a. intersect
c. parallel
b. coincide
____ 19. D is between C and E. CE = 6x, CD = 4x + 8, and DE = 27. Find CE.
a.
b.
CE = 17.5
CE = 78
c.
d.
7
CE = 105
CE = 57
Name: ________________________
ID: A
____ 20. Find the measure of ∠BOD. Then, classify the angle as acute, right, or obtuse.
a.
b.
m∠BOD = 125°; obtuse
m∠BOD = 35°; acute
c.
d.
m∠BOD = 90°; right
m∠BOD = 160°; obtuse


→
____ 21. 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°
____ 22. Find the measure of the complement of ∠M , where m∠M = 31.1°
a. 58.9°
c. 31.1°
b. 148.9°
d. 121.1°
____ 23. Find the measure of the supplement of ∠R, where m∠R = (8z + 10)°
a. (170 − 8z)°
c. 44.5°
b. (190 − 8z)°
d. (80 − 8z)°
____ 24. M is the midpoint of AN , A has coordinates (–6, –6), and M has coordinates (1, 2). Find the coordinates of N.
1
a. (8, 10)
c. (−2 2 , −2)
b.
(–5, –4)
d.
8
1
1
(8 2 , 9 2 )
Name: ________________________
ID: A
____ 25. Write the converse, inverse, and contrapositive of the conditional statement, “If an animal is a bird, then it
has two eyes.” Find the truth value of each.
a. Converse: If an animal is not a bird, then it does not have two eyes.
The converse is false.
Inverse: If an animal does not have two eyes, then it is not a bird.
The inverse is true.
Contrapositive: If an animal is a bird, then it has two eyes.
The contrapositive is true.
b. Converse: If an animal has two eyes, then it is a bird.
The converse is false.
Inverse: If an animal is not a bird, then it does not have two eyes.
The inverse is false.
Contrapositive: If an animal does not have two eyes, then it is not a bird.
The contrapositive is true.
c. Converse: If an animal does not have two eyes, then it is not a bird.
The converse is true.
Inverse: If an animal is not a bird, then it does not have two eyes.
The inverse is true.
Contrapositive: If an animal has two eyes, then it is a bird.
The contrapositive is false.
d. Converse: All birds have two eyes.
The converse is true.
Inverse: All animals have two eyes.
The inverse is true.
Contrapositive: All birds are animals, and animals have two eyes.
The contrapositive is true.
____ 26. For the conditional statement, write the converse and a biconditional statement.
If a figure is a right triangle with sides a, b, and c, then a 2 + b 2 = c 2 .
a. Converse: If a figure is not a right triangle with sides a, b, and c, then a 2 + b 2 ≠ c 2 .
Biconditional: A figure is a right triangle with sides a, b, and c if and only if
a2 + b2 = c2.
b. Converse: If a 2 + b 2 = c 2 , then the figure is a right triangle with sides a, b, and c.
Biconditional: A figure is a right triangle with sides a, b, and c if and only if
a2 + b2 = c2.
c. Converse: If a 2 + b 2 ≠ c 2 , then the figure is not a right triangle with sides a, b, and c.
Biconditional: A figure is not a right triangle with sides a, b, and c if and only if
a2 + b2 ≠ c2
d. Converse: If a 2 + b 2 ≠ c 2 , then the figure is not a right triangle with sides a, b, and c.
Biconditional: A figure is a right triangle with sides a, b, and c if and only if
a2 + b2 = c2.
9
Name: ________________________
ID: A
____ 27. 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.
____ 28. Give an example of corresponding angles.
a.
b.
∠8 and ∠4
∠4 and ∠1
c.
d.
10
∠3 and ∠6
∠5 and ∠7
Name: ________________________
ID: A
____ 29. Use the information m∠1 = (3x + 30)°, m∠2 = (5x − 10)°, and x = 20 , and the theorems you have learned to
show that l Ä m.
a.
b.
c.
d.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
By the Substitution Property of Equality, m∠1 = m∠2 = 90° .
By the Converse of the Alternate Interior Angles Theorem, l Ä m.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
Since ∠1 and ∠2 are alternate interior angles, m∠1 = m∠2 = 180° .
By the Converse of the Same-Side Interior Angles Theorem, l Ä m.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
Since ∠1 and ∠2 are same-side interior angles, m∠1 = m∠2 = 180° .
By the Converse of the Same-Side Interior Angles Theorem, l Ä m.
Since ∠1 and ∠2 are same-side interior angles, m∠1 = 3(20) + 30 = 90° and
m∠2 = 5(20) − 10 = 90° .
By substitution, m∠1 = m∠2 = 90° .
By the Converse of the Alternate Interior Angles Theorem, l Ä m.
____ 30. AB Ä CD for A(4, − 5), B(−2, − 3) , C(x, − 2), and D(6, y) . Find a set of possible values for x and y.
ÏÔÔ
¸ÔÔ
ÔÏ
Ô¸
|
1
a. ÌÔ ÊÁË x, y ˆ˜¯ | y = 3 x − 4 , x ≠ 6 ˝Ô
c. ÌÔ ÊÁË x, y ˆ˜¯ || y = 3x − 20 , y ≠ −2 ˝Ô
ÔÓ
Ô˛
Ó
˛
ÏÔÔ
¸ÔÔ
Ï
Ô
Ô¸
|
1
b. ÌÔ ÊÁË x, y ˆ˜¯ | y = 3 x − 4 ˝Ô
d. ÌÔ ÊÁË x, y ˆ˜¯ || y = 3x − 20 , x ≠ −2 ˝Ô
ÔÓ
Ô˛
Ó
˛
11
ID: A
Geometry Short Cycle 1 Exam Review
Answer Section
MULTIPLE CHOICE
1. ANS: C
A plane can be described by any three noncollinear points. Of the choices given, only points W, R, and T are
←
→
noncollinear. Thus, AC lies in plane WRT.
Feedback
A
B
C
D
Points A, C, and R are collinear. A plane can be described by any three noncollinear
points.
Points W, C, and T are collinear. A plane can be described by any three noncollinear
points.
Correct!
A plane can be described by any three noncollinear points.
PTS:
OBJ:
LOC:
TOP:
MSC:
2. ANS:
BC
1
DIF: Basic
REF: 192eb53a-4683-11df-9c7d-001185f0d2ea
1-1.3 Identifying Points and Lines in a Plane
MTH.C.11.01.05.002 | MTH.C.11.01.05.005
1-1 Understanding Points, Lines, and Planes
KEY: point | line | plane
DOK 1
C
= || −8 − (−1) ||
= |−8 + 1|
= |−7|
=7
Feedback
A
B
C
D
The length of a segment is always positive.
Find the absolute value of the difference of the coordinates.
Correct!
Find the absolute value of the difference of the coordinates.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Basic
REF: 19313ea6-4683-11df-9c7d-001185f0d2ea
1-2.1 Finding the Length of a Segment
LOC: MTH.C.11.01.02.01.001
1-2 Measuring and Constructing Segments
KEY: segment | length
DOK 1
1
ID: A
3. ANS: B
∠BOA is another name for ∠AOB, ∠BOC is another name for ∠COB, and ∠COA is another name for
∠AOC . Thus the diagram contains three angles.
Feedback
A
B
C
D
What is the name for the angle that describes the change in position from point A to
point C?
Correct!
Angle BOA is another name for angle AOB, and angle BOC is another name for angle
COB. What is the name for the angle that describes the change in position from point A
to point C?
Point O is the vertex of all the angles in the diagram.
PTS: 1
DIF: Average
REF: 193aa106-4683-11df-9c7d-001185f0d2ea
OBJ: 1-3.1 Naming Angles
LOC: MTH.C.11.02.01.002
TOP: 1-3 Measuring and Constructing Angles
KEY: angle
MSC: DOK 1
4. ANS: B
The vertical angle pairs are ∠JLK and ∠MLN, and ∠JLM and ∠KLN . These angles appear to have the same
measure.
Feedback
A
B
C
D
These angles are adjacent, not vertical.
Correct!
Vertical angles share a common vertex, the point of intersection of the two lines. The
vertex is the middle letter in the angle's name.
These angles are adjacent, not vertical.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Basic
1-4.5 Identifying Vertical Angles
1-4 Pairs of Angles
DOK 1
REF: 194b518a-4683-11df-9c7d-001185f0d2ea
LOC: MTH.C.11.02.04.06.002
KEY: vertical angles
1
ID: A
5. ANS: A
Set up the yard on a coordinate plane so that the apple tree A is at the origin, the fig tree F has coordinates
(30, 0), the plum tree P has coordinates (30, 30), and the nectarine tree N has coordinates (0, 30).
The distance between the apple tree and the plum tree is AP.
ÊÁ x − x ˆ˜ 2 + ÊÁ y − y ˆ˜ 2 =
1¯
1¯
Ë 2
Ë 2
AP =
(30 − 0) + (30 − 0) =
2
2
30 2 + 30 2 =
900 + 900 =
1800 ≈ 42.4 ft
Feedback
A
B
C
D
Correct!
Check your calculations and rounding.
Set up the yard on a coordinate plane so that the apple tree A is at the origin. Then use
the distance formula to find the distance.
Set up the yard on a coordinate plane so that the apple tree A is at the origin. Then use
the distance formula to find the distance.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Average
REF: 19599fb2-4683-11df-9c7d-001185f0d2ea
1-6.5 Application
LOC: MTH.C.11.05.04.008
1-6 Midpoint and Distance in the Coordinate Plane
KEY: coordinate geometry | distance
DOK 1
3
ID: A
6. ANS: A
The transformation is a 90° rotation with center of rotation at point O.
To be a reflection, each point and its image are the same distance from a line of reflection.
To be a translation, each point of ∆ABC moves the same distance in the same direction.
Feedback
A
B
C
D
Correct!
What happens to one of the segments in the triangle? Is B'C' an image of BC after a
rotation of 45 degrees?
The transformation is not a reflection because each point and its image are not the same
distance from a line of reflection.
The transformation is not a translation because each point of the triangle ABC does not
move the same distance in the same direction.
PTS: 1
DIF: Average
REF: 195c020e-4683-11df-9c7d-001185f0d2ea
OBJ: 1-7.1 Identifying Transformations STA: OH.OHACS.MTH.01.10.3.7
LOC: MTH.C.11.08.05.04.04.003
TOP: 1-7 Transformations in the Coordinate Plane
KEY: coordinate geometry | transformation | rotation
MSC: DOK 1
7. ANS: D
Collinear points are points that lie on the same line.
R, G, and N are three collinear points.
Feedback
A
B
C
D
Collinear points are points that lie on the same line.
Collinear points are points that lie on the same line.
Points R, P, and G are noncollinear.
Correct!
PTS: 1
DIF: Basic
REF: 19bb877e-4683-11df-9c7d-001185f0d2ea
OBJ: 1-1.1 Naming Points, Lines, and Planes
LOC: MTH.C.11.01.01.01.002
TOP: 1-1 Understanding Points, Lines, and Planes
MSC: DOK 1
4
ID: A
8. ANS: B
Conditional: If a figure is a square, then it is a rectangle.
True.
Converse: If a figure is a rectangle, then it is a square.
False. A rectangle does not necessarily have four congruent sides.
Because the converse is false, the biconditional is false.
Feedback
A
B
C
D
For a biconditional statement to be true, both the conditional statement and its converse
must be true.
Correct!
All rectangles have four 90-degree angles as well.
A rectangle does have four 90-degree angles, but does it have four congruent sides?
PTS: 1
DIF: Basic
REF: 19d823ce-4683-11df-9c7d-001185f0d2ea
OBJ: 2-4.3 Analyzing the Truth Value of a Biconditional Statement
LOC: MTH.P.08.01.04.002 | MTH.P.08.02.02.02.006
TOP: 2-4 Biconditional Statements and Definitions
KEY: biconditional | truth value
MSC: DOK 1
9. ANS: B
4x − 6 = 34
Given equation
[1] Addition Property of Equality
+6 +6
4x
= 40
Simplify.
4x
40
[2] Division Property of Equality
=
4
4
x = 10
Simplify.
Feedback
A
B
C
D
Check the properties.
Correct!
Check the properties.
Check the properties.
PTS:
OBJ:
LOC:
KEY:
1
DIF: Basic
REF: 19dce886-4683-11df-9c7d-001185f0d2ea
2-5.1 Solving an Equation in Algebra
NAT: NT.CCSS.MTH.10.9-12.A.REI.1
MTH.P.08.02.002 | MTH.C.10.06.01.01.009
TOP: 2-5 Algebraic Proof
algebraic proof | proof
MSC: DOK 2
5
ID: A
10. ANS: B
m∠JKL = m∠JKM + m∠MKL
100° = (6x + 8)° + (2x − 4)°
100 = 8x + 4
96 = 8x
12 = x
x = 12
[1] Angle Addition Postulate
Substitution Property of Equality
Simplify.
Subtraction Property of Equality
[2] Division Property of Equality
Symmetric Property of Equality
Feedback
A
B
C
D
Check the properties.
Correct!
Check the justifications.
The Segment Addition Postulate refers to segments, not angles.
PTS: 1
DIF: Average
REF: 19e1862e-4683-11df-9c7d-001185f0d2ea
OBJ: 2-5.3 Solving an Equation in Geometry
NAT: NT.CCSS.MTH.10.9-12.A.REI.1
LOC: MTH.P.08.02.001 | MTH.C.10.06.01.01.009 | MTH.C.11.02.01.01.005
TOP: 2-5 Algebraic Proof
KEY: algebraic proof | proof
MSC: DOK 2
11. ANS: D
The Transitive Property of Congruence states that if figure A ≅ figure B and figure B ≅ figure C, then figure
A ≅ figure C.
Feedback
A
B
C
D
The Reflexive Property of Congruence states that figure A is congruent to figure A.
The Substitution Property of Equality states that if a = b, then b can be substituted for a
in any expression.
The Symmetric Property of Congruence states that if figure A is congruent to figure B,
then figure B is congruent to figure A.
Correct!
PTS:
OBJ:
TOP:
MSC:
1
DIF: Basic
REF: 19e1ad3e-4683-11df-9c7d-001185f0d2ea
2-5.4 Identifying Properties of Equality and Congruence LOC: MTH.P.08.02.001
2-5 Algebraic Proof
KEY: congruence properties | reflexive | symmetric | transitive
DOK 1
6
ID: A
12. ANS: B
Step 1 Create an equation
The angles are supplements and their sum equals 180°.
(2x 2 + 3x − 5) + (x 2 + 11x − 7) = 180
Step 2 Solve the equation
3x 2 + 14x − 12 = 180
3x 2 + 14x − 192 = 0
(3x + 32)(x − 6) = 0
x = − 3 or 6 .
32
When x = − 3 , the measurement of the second angle is
32
x 2 + 11x − 7 = −10.6° .
Angles cannot have negative measurements, so x = 6 .
Step 3 Solve for the required values
The measurement of the first angle is 2x 2 + 3x − 5 = 2(6) 2 + 3(6) − 5 = 85°.
The measurement of the second angle is x 2 + 11x − 7 = (6) 2 + 11(6) − 7= 95°.
Feedback
A
B
C
D
The angles are supplements. Use the definition of supplements to solve for x.
Correct!
Check for algebra mistakes. When x equals 5, the second angle is not 120 degrees.
The angles are supplements. Use the definition of supplements to solve for x.
PTS: 1
DIF: Advanced
TOP: 2-6 Geometric Proof
MSC: DOK 2
REF: 19e8ad42-4683-11df-9c7d-001185f0d2ea
KEY: supplementary angles
7
ID: A
13. ANS: B
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. Angle Addition Postulate
4. Substitution
5. Definition of congruent angles
6. Substitution
7. Definition of complementary angles
Feedback
A
B
C
D
In a paragraph proof, statements and reasons appear together.
Correct!
In a paragraph proof, statements and reasons appear together.
In a paragraph proof, statements and reasons appear together.
PTS: 1
DIF: Average
REF: 19ed71fa-4683-11df-9c7d-001185f0d2ea
OBJ: 2-7.3 Reading a Paragraph Proof
LOC: MTH.P.08.02.03.01.002 | MTH.P.08.02.03.01.003
TOP: 2-7 Flowchart and Paragraph Proofs
KEY: paragraph proof | two column proof
MSC: DOK 1
14. ANS: A
To determine which line is the transversal for a given angle pair, locate the line that connects the vertices.
Corresponding angles lie on the same side of the transversal l, on the same sides of lines n and m.
Feedback
A
B
C
D
Correct!
Alternate interior angles lie on opposite sides of the transversal, between two lines.
To find which line is the transversal for a given angle pair, locate the line that connects
the vertices.
To find which line is the transversal for a given angle pair, locate the line that connects
the vertices.
PTS:
OBJ:
LOC:
KEY:
1
DIF: Average
REF: 1a21e5e2-4683-11df-9c7d-001185f0d2ea
3-1.3 Identifying Angle Pairs and Transversals
MTH.C.11.01.03.03.007 | MTH.C.11.01.03.03.01.001
TOP: 3-1 Lines and Angles
corresponding angles | transversal MSC: DOK 1
8
ID: A
15. ANS: C
(x)° = (3x − 70)°
0 = 2x − 70
70 = 2x
35 = x
m∠ABC = 3x − 70
m∠ABC = 3(35) − 70 = 35°
Corresponding Angles Postulate
Subtract x from both sides.
Add 70 to both sides.
Divide both sides by 2.
Substitute 35 for x. Simplify.
Feedback
A
B
C
D
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are
congruent.
Use the Corresponding Angles Postulate.
Correct!
First, set the measures of the corresponding angles equal to each other. Then, solve for
x and substitute in the expression (3x – 70).
PTS: 1
DIF: Average
REF: 1a24483e-4683-11df-9c7d-001185f0d2ea
OBJ: 3-2.1 Using the Corresponding Angles Postulate
LOC: MTH.C.11.01.03.03.01.005
TOP: 3-2 Angles Formed by Parallel Lines and Transversals
KEY: corresponding angles | parallel lines
MSC: DOK 2
16. ANS: B
Substitute 10 for x in each expression:
3x + 4 = 3(10) + 4 = 34°
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the Converse of
the Alternate Interior Angles Theorem.
Feedback
A
B
C
D
The lanes are parallel by the Converse of the Alternate Interior Angles Theorem.
Correct!
The angles are alternate interior angles.
The angles are alternate interior angles.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Average
3-3.4 Application
3-3 Proving Lines Parallel
DOK 2
REF: 1a30340a-4683-11df-9c7d-001185f0d2ea
LOC: MTH.C.11.01.03.03.01.010 | MTH.C.11.01.03.03.010
KEY: parallel lines | proof
9
ID: A
17. ANS: A
←

→
slope of AB =
←
→
slope of CD =
3 − −2
5 −7
=
5
−2
= −2
6 − 10
15 − 5
=
−4
10
= −5
5
2
The lines have different slopes, so they are not parallel.
5
2
The product of the slopes is − 2 ⋅ − 5 = 1, not −1, so the slopes are not perpendicular.
The lines are coplanar, so they cannot be skew.
Feedback
A
B
C
Correct!
The product of the slopes is 1, not –1. So the slopes are not perpendicular.
The slopes are different so they are not parallel.
PTS:
OBJ:
NAT:
TOP:
MSC:
1
DIF: Average
REF: 1a39e48a-4683-11df-9c7d-001185f0d2ea
3-5.3 Determining Whether Lines are Parallel, Perpendicular, or Neither
NT.CCSS.MTH.10.9-12.G.GPE.5 LOC: MTH.C.10.07.02.01.01.010 | MTH.C.10.07.02.01.01.013
3-5 Slopes of Lines
KEY: slope | perpendicular | parallel
DOK 1
10
ID: A
18. ANS: A
Solve the first equation for y to find the slope-intercept form. Compare the slopes and y-intercepts of both
equations.
12x + 3y = 3
3y = −12x + 3
y = −4x + 1
The slope of the first equation is –4 and the
y-intercept is 1.
y = 4x + 1
The slope of the second equation is 4 and the
y-intercept is 1.
The lines have different slopes, so they intersect.
Feedback
A
B
C
Correct!
Write both lines in slope-intercept form and compare.
Write both lines in slope-intercept form and compare.
PTS: 1
DIF: Average
REF: 1a40e48e-4683-11df-9c7d-001185f0d2ea
OBJ: 3-6.3 Classifying Pairs of Lines
LOC: MTH.C.10.07.02.01.01.006 | MTH.C.11.01.03.02.001
TOP: 3-6 Lines in the Coordinate Plane KEY: parallel | perpendicular | coordinate geometry
MSC: DOK 2
19. ANS: C
CE = CD + DE
Segment Addition Postulate
6x = (4x + 8) + 27
Substitute 6x for CE and 4x + 8 for CD.
6x = 4x + 35
Simplify.
2x = 35
Subtract 4x from both sides.
2x 35
=
Divide both sides by 2.
2
2
35
x=
or 17.5
Simplify.
2
CE = 6x = 6 (17.5) = 105
Feedback
A
B
C
D
You found the value of x. Find the length of the specified segment.
You found the length of a different segment.
Correct!
Check your equation. Make sure you are not subtracting instead of adding.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Average
REF: 1935dc4e-4683-11df-9c7d-001185f0d2ea
1-2.3 Using the Segment Addition Postulate
LOC: MTH.C.11.01.02.01.003
1-2 Measuring and Constructing Segments
KEY: segment addition postulate
DOK 2
11
ID: A
20. ANS: C
By the Protractor Postulate, m∠BOD = m∠AOD − m∠AOB .
First, measure ∠AOD and ∠AOB.
m∠BOD = m∠AOD − m∠AOB = 125° − 35° = 90°
Thus, ∠BOD is a right angle.
Feedback
A
B
C
D
To find the measure of angle BOD, subtract the measure of angle AOB from the
measure of angle AOD.
The sum of the measure of angle AOB and the measure of angle BOD is equal to the
measure of angle AOD.
Correct!
Use the Protractor Postulate.
PTS: 1
DIF: Average
REF: 193ac816-4683-11df-9c7d-001185f0d2ea
OBJ: 1-3.2 Measuring and Classifying Angles
LOC: MTH.C.11.02.01.01.001 | MTH.C.11.02.04.02.002
TOP: 1-3 Measuring and Constructing Angles
KEY: angle | measure | protractor | degrees
MSC: DOK 1
21. ANS: D
Step 1 Solve for x.
m∠ABD = m∠DBC
Definition of angle bisector.
(7x − 1)° = (4x + 8)°
Substitute 7x − 1 for ∠ABD and 4x + 8 for ∠DBC .
7x = 4x + 9
3x = 9
x=3
Add 1 to both sides.
Subtract 4x from both sides.
Divide both sides by 3.
Step 2 Find m∠ABD.
m∠ABD = 7x − 1 = 7(3) − 1 = 20°
Feedback
A
B
C
D
Check your simplification technique.
Substitute this value of x into the expression for the angle.
This answer is the entire angle. Divide by two.
Correct!
PTS:
OBJ:
LOC:
TOP:
KEY:
1
DIF: Average
REF: 193f65be-4683-11df-9c7d-001185f0d2ea
1-3.4 Finding the Measure of an Angle
MTH.C.11.02.01.01.007 | MTH.C.11.02.03.001
1-3 Measuring and Constructing Angles
angle | bisector | angle addition postulate
MSC: DOK 2
12
ID: A
22. ANS: A
Subtract from 90° and simplify.
90° − 31.1°= 58.9°
Feedback
A
B
C
D
Correct!
Find the measure of a complementary angle, not a supplementary angle.
Complementary angles are angles whose measures have a sum of 90 degrees.
The measures of complementary angles add to 90 degrees.
PTS: 1
DIF: Basic
REF: 19445186-4683-11df-9c7d-001185f0d2ea
OBJ: 1-4.2 Finding the Measures of Complements and Supplements
LOC: MTH.C.11.02.04.08.004
TOP: 1-4 Pairs of Angles
KEY: complement | complementary angles
MSC: DOK 1
23. ANS: A
Subtract from 180° and simplify.
180° − (8z + 10)° = 180 − 8z − 10 = (170 − 8z)°
Feedback
A
B
C
D
Correct!
The measures of supplementary angles add to 180 degrees.
Supplementary angles are angles whose measures have a sum of 180 degrees.
Find the measure of a supplementary angle, not a complementary angle.
PTS:
OBJ:
LOC:
KEY:
1
DIF: Average
REF: 19468cd2-4683-11df-9c7d-001185f0d2ea
1-4.2 Finding the Measures of Complements and Supplements
MTH.C.11.02.04.09.004
TOP: 1-4 Pairs of Angles
supplement | supplementary angles MSC: DOK 2
13
ID: A
24. ANS: A
Step 1 Let the coordinates of N equal (x, y).
Step 2 Use the Midpoint Formula.
ÊÁ x + x
y 1 + y 2 ˆ˜˜˜ ÊÁÁ −6 + x −6 + y ˆ˜˜
2
ÊÁ 1, 2 ˆ˜ = ÁÁÁÁ 1
˜˜
˜ = ÁÁ
,
Ë
¯ ÁÁ 2 ,
2 ˜˜
2 ˜˜˜ ÁÁ 2
¯
Ë
¯ Ë
Step 3 Find the x- and y-coordinates.
−6 + y
−6 + x
1=
Set the coordinates equal.
2=
2
2
ÊÁ −6 + x ˆ˜
ÁÊÁ −6 + y ˜ˆ˜
˜˜
ÁÁ
˜
2 (1) = 2 ÁÁÁÁ
2
(
2
)
=
2
Multiply both sides by 2.
˜
ÁÁ 2 ˜˜˜
˜
Ë 2 ¯
Ë
¯
2 = −6 + x
x=8
4 = −6 + y
y = 10
Simplify.
Solve for x or y, as appropriate.
The coordinates of N are (8, 10).
Feedback
A
B
C
D
Correct!
Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula
to solve for x and y.
This is the midpoint of line segment AM. If M is the midpoint of line segment AN, what
are the coordinates of N?
Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula
to solve for x and y.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Average
REF: 1954dafa-4683-11df-9c7d-001185f0d2ea
1-6.2 Finding the Coordinates of an Endpoint
LOC: MTH.C.11.05.04.003
1-6 Midpoint and Distance in the Coordinate Plane
KEY: coordinate geometry | midpoint
DOK 2
14
ID: A
25. ANS: B
Conditional:
Converse:
Inverse:
Contrapositive:
p→q
q→p
∼p→ ∼q
∼q→ ∼p
If an animal is a bird, then it has two eyes.
If an animal has two eyes, then it is a bird.
If an animal is not a bird, then it does not have two eyes.
If an animal does not have two eyes, then it is not a bird.
Given that the conditional statement is true, the contrapositive will also be true because the two are logically
equivalent. It is easy to find a counterexample for the converse, and since the converse and inverse are
logically equivalent, they will both be false.
Feedback
A
B
C
D
To find the contrapositive, exchange and negate the hypothesis and the conclusion.
Correct!
To find the inverse, negate the hypothesis and the conclusion.
To find the converse, exchange the hypothesis and the conclusion.
PTS: 1
DIF: Average
REF: 19cc10f2-4683-11df-9c7d-001185f0d2ea
OBJ: 2-2.4 Application
LOC: MTH.P.08.02.02.01.01.002 | MTH.P.08.02.02.01.02.002 | MTH.P.08.02.02.01.03.002
TOP: 2-2 Conditional Statements
KEY: inverse | converse | contrapositive | conditional
MSC: DOK 2
26. ANS: B
Let p and q represent the following.
p: It is a right triangle.
q: a 2 + b 2 = c 2 .
The given conditional is p → q.
The converse is q → p. If a 2 + b 2 = c 2 , then the figure is a right triangle with sides a, b, and c.
The biconditional is p ↔ q. A figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2 .
Feedback
A
B
C
D
Find the converse, not inverse.
Correct!
Find the converse, not the contrapositive.
Find the converse, not the contrapositive.
PTS:
OBJ:
LOC:
TOP:
MSC:
1
DIF: Average
REF: 19d7fcbe-4683-11df-9c7d-001185f0d2ea
2-4.2 Writing a Biconditional Statement
MTH.P.08.02.02.01.01.002 | MTH.P.08.02.02.02.004
2-4 Biconditional Statements and Definitions
KEY: biconditional
DOK 2
15
ID: A
27. ANS: A
P = 2l + 2w
26 = 2(8) + 2w
26 = 16 + 2w
−16 = −16
10 = 2w
10 2w
=
2
2
5=w
w=5
Given equation
[1] Substitution Property of Equality
Simplify.
Subtraction Property of Equality
Simplify.
[2] Division Property of Equality
Simplify.
Symmetric Property of Equality
Feedback
A
B
C
D
Correct!
The variables P and l are substituted, not simplified. Use the Substitution Property.
Check the properties.
Check the justifications.
PTS: 1
DIF: Average
REF: 19df23d2-4683-11df-9c7d-001185f0d2ea
OBJ: 2-5.2 Problem-Solving Application
NAT: NT.CCSS.MTH.10.9-12.A.REI.1
LOC: MTH.P.08.02.002 | MTH.C.10.06.01.01.009
TOP: 2-5 Algebraic Proof
KEY: algebraic proof | proof
MSC: DOK 2
28. ANS: A
Corresponding angles lie on the same side of a transversal, on the same sides of the two lines the transversal
crosses. So, ∠8 and ∠4 are corresponding angles.
Feedback
A
B
C
D
Correct!
Angle 4 and angle 1 are supplementary angles, not corresponding angles.
Corresponding angles lie on the same side of a transversal, on the same sides of two
lines.
Angle 5 and angle 7 are vertical angles, not corresponding angles.
PTS:
OBJ:
TOP:
MSC:
1
DIF: Basic
3-1.2 Classifying Pairs of Angles
3-1 Lines and Angles
DOK 1
REF: 1a1f8386-4683-11df-9c7d-001185f0d2ea
LOC: MTH.C.11.01.03.03.01.001
KEY: corresponding angles | transversal
16
ID: A
29. ANS: A
m∠1 = 3(20) + 30 = 90° ;
m∠2 = 5(20) − 10 = 90°
m∠1 = m∠2 = 90°
lÄm
Substitute 20 for x.
Substitution Property of Equality
Converse of the Alternate Interior Angles Theorem
Feedback
Correct!
Angles 1 and 2 are alternate interior angles and are congruent.
Angles 1 and 2 are alternate interior angles and are congruent.
Angles 1 and 2 are alternate interior angles and are congruent.
A
B
C
D
PTS:
OBJ:
LOC:
KEY:
30. ANS:
1
DIF: Average
REF: 1a2b9662-4683-11df-9c7d-001185f0d2ea
3-3.2 Determining Whether Lines are Parallel
MTH.C.11.01.03.03.01.010 | MTH.C.11.01.03.03.010
TOP: 3-3 Proving Lines Parallel
parallel | alternate interior angles
MSC: DOK 2
A
−3 − (−5)
2
1
=
=−
slope of AB =
−2 − 4
−6
3
y − (−2) y + 2
slope of CD =
=
,x≠6
6−x
6−x
Parallel lines have the same slope. Write an equation comparing the
y+2
1
=−
6−x
3
slopes of AB and CD.
−3(y + 2) = 1(6 − x)
Cross multiply.
−3y − 6 = 6 − x
Distribute.
−3y = 12 − x
Simplify.
y = 3 x−4
1
The set of possible values for x and y is
ÔÏÔ Ê
|
1
ÌÔ ÁË x, y ˜ˆ¯ | y = 3 x − 4 , x ≠
ÔÓ
Ô¸Ô
6˝ .
Ô˛
Feedback
A
B
C
D
Correct!
Check the constraints of x.
In the slope formula, the difference of the y-values is in the numerator and the
difference of the x-values in the denominator.
First, find the slopes of the segments. Then, set the slopes equal to each other and cross
multiply and simplify.
PTS:
NAT:
LOC:
TOP:
MSC:
1
DIF: Advanced
REF: 1a3c1fd6-4683-11df-9c7d-001185f0d2ea
NT.CCSS.MTH.10.9-12.G.GPE.5 STA: OH.OHACS.MTH.01.10.4.9
MTH.C.10.07.02.01.01.004 | MTH.C.10.07.02.01.01.009
3-5 Slopes of Lines
KEY: slope | parallel
DOK 3
17