geo final review 2014 - mrzmath.com
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Name: ________________________ Period: ___________________ Date: __________
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Geo Final Review 2014
Multiple ChoiceIdentify the choice that best completes the statement or answers the question.
____ 1. An angle measures 2 degrees more than 3 times its complement. Find the measure of its complement.a. 23° c. 22°b. 272° d. 68°
____ 2. BD→
bisects ∠ABC, m∠ABD = (7x − 1)°, and m∠DBC = (4x + 8)°. Find m∠ABD.a. m∠ABD = 22° c. m∠ABD = 20°b. m∠ABD = 3° d. m∠ABD = 40°
____ 3. Use the information m∠1 = (3x + 30)°, m∠2 = (5x − 10)°, andx = 20, and the theorems you have learned to show that l Ä m.
a. 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.
b. 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.
c. 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.
d. 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.
Name: ________________________ ID: A
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____ 4. In a dance performance, four dancers form a diamond with vertices A(2, 0),B(0, 2),C(−2, 0), andD(0, − 2).
Then, they move along the dance floor following the translation vector, 0, 4 . There they pause, and then
move again along the same vector. What are their coordinates after six such translations?a. A′(26, 0),B′(24, 2),C ′(22, 0), andD ′(24, − 2)b. A′(2, 10),B′(0, 12),C ′(−2, 10), andD ′(0, 8)c. A′(2, 24),B′(0, 26),C ′(−2, 24), andD ′(0, 22)d. A′(26, 24),B′(24, 6),C ′(22, 24), andD ′(24, 22)
____ 5. Two sides of an equilateral triangle measure (2y + 3) units and (y2 − 5) units. If the perimeter of the triangle is 33 units, what is the value of y?a. y = 4 c. y = 7b. y = 11 d. y = 15
Name: ________________________ ID: A
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____ 6. Use the given paragraph proof to write a two-column proof.
Given: ∠BAC is a right angle. ∠1 ≅ ∠3Prove: ∠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 Reasons
1. ∠BAC is a right angle. ∠1 ≅ ∠3 1. Given2. m∠BAC = 90° 2. Definition of a right angle3. m∠BAC = m∠1+ m∠2 3. [1]4. m∠1+ m∠2 = 90° 4. Substitution5. m∠1 = m∠3 5. [2]6. m∠3+ m∠2 = 90° 6. Substitution 7. ∠2and∠3 are complementary. 7. Definition of complementary angles
a. [1] Angle Addition Postulate[2] Definition of congruent angles
c. [1] Substitution[2] Definition of equality
b. [1] Angle Addition Postulate[2] Definition of equality
d. [1] Substitution[2] Definition of congruent angles
____ 7. Find the measure of the complement of ∠M , where m∠M = 31.1°a. 58.9° c. 121.1°b. 148.9° d. 31.1°
____ 8. One of the acute angles in a right triangle has a measure of 34.6°. What is the measure of the other acute angle?a. 90° c. 34.6°b. 145.4° d. 55.4°
Name: ________________________ ID: A
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____ 9. Draw two lines and a transversal such that ∠1 and ∠2 are alternate interior angles, ∠2 and ∠3 are corresponding angles, and ∠3 and ∠4 are alternate exterior angles. What type of angle pair is ∠1 and ∠4?a.
∠1 and ∠4 are corresponding angles.b.
∠1 and ∠4 are supplementary angles.c.
∠1 and ∠4 are vertical angles.d.
∠1 and ∠4 are alternate exterior angles.
Name: ________________________ ID: A
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____ 10. Write a justification for each step.
m∠JKL = 100°
m∠JKL = m∠JKM + m∠MKL [1]100° = (6x + 8)° + (2x − 4)° Substitution Property of Equality100 = 8x + 4 Simplify.96 = 8x Subtraction Property of Equality12 = x [2]x = 12 Symmetric Property of Equality
a. [1] Angle Addition Postulate[2] Simplify.
b. [1] Angle Addition Postulate[2] Division Property of Equality
c. [1] Segment Addition Postulate[2] Multiplication Property of Equality
d. [1] Transitive Property of Equality[2] Division Property of Equality
____ 11. The figure shows part of the roof structure of a house. Use SAS to explain why ∆RTS ≅ ∆RTU .
Complete the explanation.
It is given that [1]. Since ∠RTS and ∠RTU are right angles, [2] by the Right Angle Congruence Theorem. By the Reflexive Property of Congruence, [3]. Therefore, ∆RTS ≅ ∆RTU by SAS.
a. [1] ST ≅ UT[2] ∠RTS ≅ ∠RTU[3] RT ≅ RT
c. [1] ST ≅ UT[2] ∠SRT ≅ ∠URT[3] ST ≅ UT
b. [1] ST ≅ UT[2] ∠RTS ≅ ∠RTU[3] SU ≅ SU
d. [1] RT ≅ RT[2] ∠SRT ≅ ∠URT[3] ST ≅ UT
Name: ________________________ ID: A
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____ 12. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
a. polygon, decagon c. polygon, dodecagonb. polygon, hexagon d. not a polygon
____ 13. Use the Converse of the Corresponding Angles Postulate and ∠1 ≅ ∠2 to show that l Ä m.
a. By the Converse of the Corresponding Angles Postulate, ∠1 ≅ ∠2. From the diagram, l Ä m.
b. ∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are alternate interior angles. So by the Converse of the Alternate Interior Angles Postulate, l Ä m.
c. ∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the Corresponding Angles Postulate, l Ä m.
d. ∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the Converse of the Corresponding Angles Postulate, l Ä m.
____ 14. Find m∠DCB, given ∠A ≅ ∠F, ∠B ≅ ∠E, and m∠CDE = 46°.
a. m∠DCB = 44° c. m∠DCB = 67°b. m∠DCB = 134° d. m∠DCB = 46°
Name: ________________________ ID: A
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____ 15. Determine if you can use ASA to prove ∆CBA ≅ ∆CED. Explain.
a. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem, ∠ACB ≅ ∠DCE. Therefore, ∆CBA ≅ ∆CED by ASA.
b. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Adjacent Angles Theorem, ∠ACB ≅ ∠DCE. Therefore, ∆CBA ≅ ∆CED by ASA.
c. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem, ∠ACB ≅ ∠DCE. Therefore, ∆CBA ≅ ∆CED by SAS.
d. AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. No other congruence relationships can be determined, so ASA cannot be applied.
____ 16. Translate the triangle with vertices A(3, 4),B(2, − 1), andC(4, 12) along the vector −1, 3 . Find the
coordinates of the new image.a. A′(2, 7),B′(1, 2), andC ′(3, 15)b. A′(4, 7),B′(3, − 2), andC ′(5, 15)c. A′(6, 3),B′(5,− 2), andC ′(7, 11)d. A′(−3, 12),B′(−2, − 3), andC ′(−4, 36)
Name: ________________________ ID: A
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____ 17. Given: ∠MLN ≅ ∠PLO, ∠MNL ≅ ∠POL, MO ≅ NPProve: ∆MLP is isosceles.
Complete the proof.
Proof:Statements Reasons
1. ∠MLN ≅ ∠PLO, ∠MNL ≅ ∠POL 1. Given
2. MO ≅ NP 2. Given
3. MO = NP 3. Definition of congruent line segments4. NO = NO 4. Reflexive Property of Equality5. MO − NO = NP − NO 5. Subtraction Property of Equality6. MO − NO = MN and NP − NO = OP 6. Segment Addition Postulate7. MN = OP 7. Substitution Property of Equality8. ∆MLN ≅ ∆PLO 8. [1]
9. ML ≅ PL 9. [2]
10. ∆MLP is isosceles. 10. Definition of isosceles triangle
a. [1] ASA[2] CPCTC
c. [1] AAS[2] CPCTC
b. [1] CPCTC[2] AAS
d. [1] CPCTC[2] ASA
Name: ________________________ ID: A
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____ 18. 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. 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.
b. 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.
c. 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.
d. 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.
Name: ________________________ ID: A
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____ 19. Find CA.
a. CA = 10b. Not enough information. An equiangular triangle is not necessarily equilateral.c. CA = 12d. CA = 14
Name: ________________________ ID: A
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____ 20. 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 Reasons
1. ∠1 and ∠2 are supplementary. 1. Given2. [1] 2. Given3. m∠1 + m∠2 = 180° 3. [2]4. 135° + m∠2 = 180° 4. Substitution Property5. m∠2 = 45° 5. [3]
a. [1] m∠1 = 135°[2] Definition of supplementary angles[3] Subtraction Property of Equality
b. [1] m∠1 = 135°[2] Definition of supplementary angles[3] Substitution Property
c. [1] m∠1 = 135°[2] Definition of complementary angles[3] Subtraction Property of Equality
d. [1] m∠2 = 135°[2] Definition of supplementary angles[3] Subtraction Property of Equality
Name: ________________________ ID: A
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____ 21. Write a justification for each step, given that EG = FH .
EG = FH Given informationEG = EF + FG [1]FH = FG + GH Segment Addition PostulateEF + FG = FG + GH [2]EF = GH Subtraction Property of Equality
a. [1] Substitution Property of Equality[2] Transitive Property of Equality
b. [1] Segment Addition Postulate[2] Definition of congruent segments
c. [1] Segment Addition Postulate[2] Substitution Property of Equality
d. [1] Angle Addition Postulate[2] Subtraction Property of Equality
Name: ________________________ ID: A
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____ 22. Given: RT ⊥SU , ∠SRT ≅ ∠URT , RS ≅ RU . T is the midpoint of SU .
Prove: ∆RTS ≅ ∆RTU
Complete the proof.
Proof:Statements Reasons
1. RT ⊥SU 1. Given
2. ∠RTS and ∠RTU are right angles. 2. [1]3. ∠RTS ≅ ∠RTU 3. Right Angle Congruence Theorem4. ∠SRT ≅ ∠URT 4. Given5. ∠S ≅ ∠U 5. [2]
6. RS ≅ RU 6. Given
7. T is the midpoint of SU . 7. Given
8. ST ≅ UT 8. Definition of midpoint
9. RT ≅ RT 9. [3]
10. ∆RTS ≅ ∆RTU 10. Definition of congruent triangles
a. [1] Definition of perpendicular lines[2] Third Angles Theorem[3] Reflexive Property of Congruence
b. [1] Definition of perpendicular lines[2] Third Angles Theorem[3] Symmetric Property of Congruence
c. [1] Definition of right angles[2] Third Angles Theorem[3] Transitive Property of Congruence
d. [1] Definition of perpendicular lines[2] Vertical Angles Theorem[3] Symmetric Property of Congruence
Name: ________________________ ID: A
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____ 23. Tell whether the transformation appears to be a reflection. Explain.
a. No; the image does not appear to be flipped.b. Yes; the image appears to be flipped across a line.
____ 24. Find m∠RST .
a. m∠RST = 156° c. m∠RST = 108°b. m∠RST = 24° d. m∠RST = 72°
____ 25. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a. m∠1 = 80° c. m∠1 = 75°b. m∠1 = 95° d. m∠1 = 85°
Name: ________________________ ID: A
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____ 26. A billiard ball bounces off the sides of a rectangular billiards table in such a way that ∠1 ≅ ∠3, ∠4 ≅ ∠6, and ∠3 and ∠4 are complementary. If m∠1 = 26.5°, find m∠3, m∠4, and m∠5.
a. m∠3 = 63.5°; m∠4 = 26.5°; m∠5 = 53°b. m∠3 = 26.5°; m∠4 = 153.5°; m∠5 = 26.5°c. m∠3 = 26.5°; m∠4 = 63.5°; m∠5 = 63.5°d. m∠3 = 26.5°; m∠4 = 63.5°; m∠5 = 53°
Name: ________________________ ID: A
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____ 27. Use the given two-column proof to write a flowchart proof.Given: ∠1 ≅ ∠4Prove: m∠2 = m∠3
Two-column proof:Statements Reasons
1. ∠1 ≅ ∠4 1. Given2. ∠1 and ∠2 are supplementary. ∠3 and ∠4 are supplementary.
2. Definition of linear pair
3. ∠2 ≅ ∠3 3. Congruent Supplements Theorem4. m∠2 = m∠3 4. Definition of congruent segments
Complete the proof.
Flowchart proof:∠1 ≅ ∠4
Given
[1] ∠2 ≅ ∠3 m∠2 = m∠3
Definition of linear pair [2] Definition of congruent segments
a. [1] ∠2 ≅ ∠3[2] Definition of congruent segments
b. [1] ∠1 and ∠2 are supplements; ∠3 and ∠4 are supplementary[2] Congruent Complements Theorem
c. [1] ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary[2] Congruent Supplements Theorem
d. [1] Definition of congruent segments[2] Congruent Supplements Theorem
Name: ________________________ ID: A
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____ 28. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a. m∠1 = 120° c. m∠1 = 125°b. m∠1 = 130° d. m∠1 = 135°
____ 29. The point G(4, 8) is rotated 90° about point M(−7, − 9) and then reflected across the line y = −6. Find the coordinates of the image G ′.a. (−24, − 14) c. (−18, − 20)b. (12, − 14) d. (−8, − 16)
Name: ________________________ ID: A
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____ 30. Given: P is the midpoint of TQ and RS .Prove: ∆TPR ≅ ∆QPS
Complete the proof.
Proof:Statements Reasons
1. P is the midpoint of TQ and RS . 1. Given
2. TP ≅ QP, RP ≅ SP 2. [1]
3. [2] 3. Vertical Angles Theorem4. ∆TPR ≅ ∆QPS 4. [3]
a. [1] Definition of midpoint
[2] RT ≅ SQ[3] SSS
c. [1] Definition of midpoint[2] ∠TPR ≅ ∠QPS[3] SSS
b. [1]. Definition of midpoint[2] ∠TPR ≅ ∠QPS[3] SAS
d. [1] Definition of midpoint[2] ∠PRT ≅ ∠PSQ[3] SAS
____ 31. Draw and label a pair of opposite rays FG→
and FH→
.a. c.
b. d.
Name: ________________________ ID: A
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____ 32. Using the information about John, Jason, and Julie, can you uniquely determine how they stand with respect to each other? On what basis?
Statement 1: John and Jason are standing 12 feet apart.Statement 2: The angle from Julie to John to Jason measures 31º.Statement 3: The angle from John to Jason to Julie measures 49º.
a. Yes. They form a unique triangle by SAS.b. No. There is no unique configuration.c. Yes. They form a unique triangle by ASA.d. Yes. They form a unique triangle by SSS.
____ 33. Find the measure of each exterior angle of a regular decagon.a. 18° c. 22.5°b. 36° d. 45°
____ 34. Tell whether the transformation appears to be a translation. Explain.
a. Yes; all of the points have moved the same distance in the same direction.b. No; not all of the points have moved the same distance.
____ 35. Give an example of corresponding angles.
a. ∠3 and ∠6 c. ∠5 and ∠7b. ∠4 and ∠1 d. ∠8 and ∠4
Name: ________________________ ID: A
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____ 36. Tell whether the transformation appears to be a rotation. Explain.
a. Yes; the figure appears to be turned around a point.b. No; the figure appears to be flipped.
Name: ________________________ ID: A
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____ 37. Given: ∠CBF ≅ ∠CDG, AC bisects ∠BADProve: AD ≅ AB
Complete the flowchart proof.
Proof:
∠CBF ≅ ∠CDG ∠ABC ≅ ∠ADC
Given 1.____________
AC bisects ∠BAD
2.____________ ∆ACB ≅ ∆ACD AD ≅ AB
Given. Definition of angle bisector.
4.___________ 5._________
AC ≅ AC
3.____________
a. 1. Congruent Supplements Theorem2. ∠CAB ≅ ∠CAD3. Reflexive Property of Congruence4. AAS5. CPCTC
c. 1. Congruent Complements Theorem2. ∠ACB ≅ ∠ACD3. Reflexive Property of Congruence4. CPCTC5. AAS
b. 1. Congruent Complements Theorem2. ∠ACB ≅ ∠ACD3. Transitive Property of Congruence4. CPCTC5. AAS
d. 1. Congruent Supplements Theorem2. ∠CAB ≅ ∠CAD3. Transitive Property of Congruence4. AAS5. CPCTC
Name: ________________________ ID: A
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____ 38. Use AAS to prove the triangles congruent.
Given: AB Ä GH , AC→←
Ä FH→←
, AC ≅ FH
Prove: ∆ABC ≅ ∆HGF
Complete the flowchart proof.
Proof:
AB Ä GH ∠B ≅ ∠G
Given 1.___________
AC→←
Ä FH→← ∠ACB ≅ ∠HFG ∆ABC ≅ ∆HGF
Given 2.___________ AAS
AC ≅ FH
Given
a. 1. Alternate Interior Angles Theorem2. Alternate Exterior Angles Theorem
b. 1. Alternate Interior Angles Theorem2. Alternate Interior Angles Theorem
c. 1. Alternate Exterior Angles Theorem2. Alternate Interior Angles Theorem
d. 1. Alternate Exterior Angles Theorem2. Alternate Exterior Angles Theorem
Name: ________________________ ID: A
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____ 39. Rotate ∆RSQ with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin.a. c.
b. d.
____ 40. Find the measure of each interior angle of a regular 45-gon.a. 188° c. 172°b. 176° d. 164°
Name: ________________________ ID: A
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____ 41. Write a two-column proof.Given: t ⊥ l, ∠1 ≅ ∠2Prove: m Ä l
Complete the proof.
Proof:Statements Reasons
1. [1] 1. Given2. t ⊥ m 2. [2]3. m Ä l 3. [3]
a. [1] t ⊥ l,∠1 ≅ ∠2[2] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.[3] Perpendicular Transversal Theorem
b. [1] t ⊥ l,∠1 ≅ ∠2[2] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.[3] 2 lines ⊥ to the same line → lines Ä.
c. [1] t ⊥ l,∠1 ≅ ∠2[2] 2 lines ⊥ to the same line → lines Ä.[3] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.
d. [1] t ⊥ l,∠1 ≅ ∠2[2] Perpendicular Transversal Theorem[3] 2 lines ⊥ to the same line → lines Ä.
Name: ________________________ ID: A
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____ 42. Use the given flowchart proof to write a two-column proof of the statement AF ≅ FD.
Flowchart proof:AB = CD;BF = FC
AB + BF = AF
FC + CD = FD
GivenSegment Addition Postulate
AB + BF =
FC + CDAF = FD AF ≅ FD
Addition Property of
Equality
Substitution Definition of congruent segments
Complete the proof.
Two-column proof:Statements Reasons
1. AB = CD; BF = FC 1. Given2. [1] 2. Addition Property of Equality3. [2] 3. Segment Addition Postulate4. AF = FD 4. Substitution
5. AF ≅ FD 5. Definition of congruent segments
a. [1] AB + BF = AF ; FC + CD = FD[2] AF = FD
b. [1] AB = CD; BF = FC[2] AB + BF = FC + CD
c. [1] AF = FD[2] AB + BF = FC + CD
d. [1] AB + BF = FC + CD[2] AB + BF = AF ;FC + CD = FD
Name: ________________________ ID: A
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____ 43. Identify the transversal and classify the angle pair ∠11 and ∠7.
a. The transversal is line m. The angles are corresponding angles.b. The transversal is line l. The angles are corresponding angles.c. The transversal is line l. The angles are alternate interior angles.d. The transversal is line n. The angles are alternate exterior angles.
____ 44. Show ∆ABD ≅ ∆CDB for a = 3.
Complete the proof.
AB = a + 7 = [1] = 10CD = 4a − 2 = [2] = 12− 2 = 10AD = 6a − 2 = 6(3)− 2 = 18− 2 = [3]CB = [4]
AB ≅ CD. AD ≅ CB. BD ≅ BD by the Reflexive Property of Congruence. So ∆ABD ≅ ∆CDB by [5].
a. [1] 3+ 7[2] 4(3)− 2[3] 16[4] 16[5] SSS
b. [1] a + 7[2] 4a − 2[3] 16[4] 16[5] SAS
c. [1] 3+ 7[2] 4(3)− 2[3] 26[4] 26[5] SSS
d. [1] 3+ 7[2] 4(3)− 2[3] 16[4] 16[5] SAS
Name: ________________________ ID: A
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____ 45. Tell whether ∠FAC and ∠3 are only adjacent, adjacent and form a linear pair, or not adjacent.
a. only adjacentb. not adjacentc. adjacent and form a linear pair
____ 46. Find m∠E and m∠N , given m∠F = m∠P , m∠E = (x2)°, and m∠N = (4x2 − 75)°.
a. m∠E = 65°, m∠N = 65° c. m∠E = 65°, m∠N = 25°b. m∠E = 25°, m∠N = 65° d. m∠E = 25°, m∠N = 25°
____ 47. D is between C and E. CE = 6x, CD = 4x + 8, and DE = 27. Find CE.
a. CE = 17.5 c. CE = 57b. CE = 105 d. CE = 78
Name: ________________________ ID: A
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____ 48. Tell whether the transformation appears to be a translation. Explain.
a. Yes; all of the points have moved the same distance in the same direction.b. No; not all of the points have moved the same distance.
____ 49. Find m∠ABC.
a. m∠ABC = 45º c. m∠ABC = 40ºb. m∠ABC = 50º d. m∠ABC = 35º
Name: ________________________ ID: A
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____ 50. Use the given plan to write a two-column proof.Given: m∠1 + m∠2 = 90°, m∠3 + m∠4 = 90°, m∠2 = m∠3
Prove: m∠1 = m∠4Plan: Since both pairs of angle measures add to 90°, use substitution to show that the sums of both pairs are equal. Since m∠2 = m∠3, use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m∠1 = m∠4.
Complete the proof.
Proof:Statements Reasons
1. m∠1 + m∠2 = 90° 1. Given2. [1] 2. Given3. m∠1 + m∠2 = m∠3 + m∠4 3. Substitution Property4. m∠2 = m∠3 4. Given5. m∠1 + m∠2 = m∠2 + m∠4 5. [2]6. m∠1 = m∠4 6. [3]
a. [1] m∠3 + m∠4 = 90°[2] Subtraction Property of Equality[3] Substitution Property
b. [1] m∠3 + m∠4 = 90°[2] Substitution Property[3] Subtraction Property of Equality
c. [1] m∠5 + m∠6 = 90°[2] Addition Property of Equality[3] Substitution Property
d. [1] m∠5 + m∠6 = 90°[2] Substitution Property[3] Subtraction Property of Equality
ID: A
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Geo Final Review 2014Answer Section
MULTIPLE CHOICE
1. ANS: CLet m∠A = x°. Then m∠B = (90− x)°.
m∠A = 3m∠B + 2x = 3(90− x) + 2 Substitute.x = 270− 3x + 2 Distribute.x = 272− 3x Combine like terms.4x = 272 Add 3x to both sides.
x = 272
4 Divide both sides by 4.
x = 68 Simplify.
The measure of ∠A is 68°, so its complement is 22°.
Feedback
A Check your equation. The original angle is 2 degrees more than 3 times its complement.B Simplify the terms when solving.C Correct!D This is the original angle. Find the measure of the complement.
PTS: 1 DIF: Average REF: Page 29 OBJ: 1-4.3 Using Complements and Supplements to Solve Problems NAT: 12.3.3.g STA: 6MG2.2 TOP: 1-4 Pairs of Angles KEY: complementary angles | supplementary angles
ID: A
2
2. ANS: CStep 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 Add 1 to both sides.3x = 9 Subtract 4x from both sides.x = 3 Divide both sides by 3.
Step 2 Find m∠ABD.m∠ABD = 7x − 1 = 7(3)− 1 = 20°
Feedback
A Check your simplification technique.B Substitute this value of x into the expression for the angle.C Correct!D This answer is the entire angle. Divide by two.
PTS: 1 DIF: Average REF: Page 23 OBJ: 1-3.4 Finding the Measure of an Angle NAT: 12.2.1.fSTA: GE1.0 TOP: 1-3 Measuring and Constructing Angles KEY: angle bisectors | angle measures
3. ANS: Cm∠1 = 3(20)+ 30= 90°; m∠2 = 5(20)− 10= 90° Substitute 20 for x.
m∠1 = m∠2 = 90° Substitution Property of Equalityl Ä m Converse of the Alternate Interior Angles Theorem
Feedback
A Angles 1 and 2 are alternate interior angles and are congruent.B Angles 1 and 2 are alternate interior angles and are congruent.C Correct!D Angles 1 and 2 are alternate interior angles and are congruent.
PTS: 1 DIF: Average REF: Page 164 OBJ: 3-3.2 Determining Whether Lines are Parallel NAT: 12.3.3.gSTA: GE7.0 TOP: 3-3 Proving Lines Parallel
ID: A
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4. ANS: CThe complete translation is (x, y) → (x, y) + 6 ⋅ (0, 4) = (x, y) + (0, 24)= (x + 0, y + 24) = (x, y + 24).A(2, 0)→ A(2, 0+ 24) = A′(2, 24)
B(0, 2)→ B(0, 2+ 24) = B′(0, 26)
C(−2, 0)→ C(−2, 0+ 24) = C ′(−2, 24)
D(0, − 2) → D(0, − 2+ 24) = D ′(0, 22)
Feedback
A Add 24 to each y-coordinate.B Add 24 to each y-coordinate.C Correct!D Add 24 to each y-coordinate.
PTS: 1 DIF: Average REF: Page 833 OBJ: 12-2.4 ApplicationNAT: 12.3.2.c STA: GE22.0 TOP: 12-2 Translations
5. ANS: AThe perimeter is 33 units and it is an equilateral triangle, so each side has length 11 units.Use this to solve for either side.11= 2y + 3
8 = 2y
4 = y
11= y2 − 5
16= y2
4 = yAn answer of −4 does not apply here.
Feedback
A Correct!B This is the length of each side. Now find the value of y.C When solving 2y + 3 = 11, subtract 3 from both sides of the equation.D The perimeter is 33 so the length of each side is 11. Set one of the sides equal to 11 and
solve for y.
PTS: 1 DIF: Advanced NAT: 12.2.1.h STA: GE12.0TOP: 4-1 Classifying Triangles
ID: A
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6. ANS: ATwo-column proof:
Statements Reasons1. ∠BAC is a right angle. ∠1 ≅ ∠3 1. Given2. m∠BAC = 90° 2. Definition of a right angle3. m∠BAC = m∠1+ m∠2 3. Angle Addition Postulate4. m∠1+ m∠2 = 90° 4. Substitution5. m∠1 = m∠3 5. Definition of congruent angles6. m∠3+ m∠2 = 90° 6. Substitution 7. ∠2and∠3 are complementary. 7. Definition of complementary angles
Feedback
A Correct!B In a paragraph proof, statements and reasons appear together.C In a paragraph proof, statements and reasons appear together.D In a paragraph proof, statements and reasons appear together.
PTS: 1 DIF: Average REF: Page 120 OBJ: 2-7.3 Reading a Paragraph ProofNAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs
7. ANS: ASubtract from 90º and simplify.90° − 31.1°= 58.9°
Feedback
A Correct!B Find the measure of a complementary angle, not a supplementary angle.C The measures of complementary angles add to 90 degrees.D Complementary angles are angles whose measures have a sum of 90 degrees.
PTS: 1 DIF: Basic REF: Page 29 OBJ: 1-4.2 Finding the Measures of Complements and Supplements NAT: 12.3.3.g STA: 6MG2.2 TOP: 1-4 Pairs of Angles KEY: complementary angles | supplementary angles
ID: A
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8. ANS: DLet the acute angles be ∠M and ∠N , with m∠M = 34.6°.m∠M + m∠N = 90° The acute angles of a right triangle are complementary.34.6° + m∠N = 90° Substitute 34.6° for m∠M .m∠N = 55.4° Subtract 34.6° from both sides.
Feedback
A The measure of the other acute angle is less than 90 degrees.B The two acute angles in a right triangle are complementary.C This is the measure of the given angle. Find the measure of the other acute angle.D Correct!
PTS: 1 DIF: Basic REF: Page 225 OBJ: 4-2.2 Finding Angle Measures in Right Triangles NAT: 12.3.3.fSTA: GE12.0 TOP: 4-2 Angle Relationships in Triangles
ID: A
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9. ANS: AStep 1 Draw two lines m, n, and a transversal p such that ∠1 and ∠2 are alternate interior angles. They should lie on opposite sides of the transversal p between lines m and n.
Step 2 ∠2 and ∠3 are corresponding angles. Corresponding angles lie on the same side of the transversal p and on the same sides of lines m and n. Add ∠3 to the drawing.
Step 3 ∠3 and ∠4 are alternate exterior angles. They should lie on opposite sides of the transversal p and outside lines m and n. Add ∠4 to the drawing.
∠1 and ∠4 are corresponding angles. They lie on the same side of the transversal p and on the same sides of lines m and n.
Feedback
A Correct!B Angles 2 and 3 are corresponding angles and should lie on the same side of transversal
p, on the same sides of lines m and n.C Angles 2 and 3 are corresponding angles and should lie on the same side of transversal
p, on the same sides of lines m and n.D Angles 1 and 2 are alternate interior angles and should lie on opposite sides of
transversal p, between lines m and n.
PTS: 1 DIF: Advanced NAT: 12.2.1.f STA: GE7.0TOP: 3-1 Lines and Angles KEY: multi-step
ID: A
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10. ANS: Bm∠JKL = m∠JKM + m∠MKL [1] Angle Addition Postulate100° = (6x + 8)° + (2x − 4)° Substitution Property of Equality100 = 8x + 4 Simplify.96 = 8x Subtraction Property of Equality12 = x [2] Division Property of Equalityx = 12 Symmetric Property of Equality
Feedback
A Check the justifications.B Correct!C The Segment Addition Postulate refers to segments, not angles.D Check the properties.
PTS: 1 DIF: Average REF: Page 106 OBJ: 2-5.3 Solving an Equation in Geometry NAT: 12.5.2.eSTA: GE1.0 TOP: 2-5 Algebraic Proof
11. ANS: A
It is given that ST ≅ UT . Since ∠RTS and ∠RTU are right angles, ∠RTS ≅ ∠RTU by the Right Angle
Congruence Theorem. By the Reflexive Property of Congruence, RT ≅ RT . Therefore, ∆RTS ≅ ∆RTU by SAS.
Feedback
A Correct!B Segment SU being congruent to itself does not help in proving the triangles congruent.C Angle SRT and angle URT are not right angles.D Check the figure to see what is given.
PTS: 1 DIF: Average REF: Page 243 OBJ: 4-4.2 ApplicationNAT: 12.3.5.a STA: GE5.0 TOP: 4-4 Triangle Congruence: SSS and SAS
12. ANS: AA polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. A polygon with 10 sides is called a decagon.
Feedback
A Correct!B A hexagon has 6 sides.C A dodecagon has 12 sides.D A polygon is a closed plane figure formed by three or more segments that intersect only
at their endpoints.
PTS: 1 DIF: Basic REF: Page 382 OBJ: 6-1.1 Identifying PolygonsNAT: 12.3.3.f STA: GE12.0 TOP: 6-1 Properties and Attributes of Polygons
ID: A
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13. ANS: D∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the Converse of the Corresponding Angles Postulate, l Ä m.
Feedback
A Use the given information.B Use the Converse of the Corresponding Angles Postulate.C Use the Converse of the Corresponding Angles Postulate.D Correct!
PTS: 1 DIF: Basic REF: Page 162 OBJ: 3-3.1 Using the converse of the Corresponding Angles Postulate NAT: 12.3.3.g STA: GE7.0 TOP: 3-3 Proving Lines Parallel
14. ANS: DThe Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent.
It is given that ∠A ≅ ∠F and ∠B ≅ ∠E. Therefore, ∠CDE ≅ ∠DCB. So, m∠DCB = 46°.
Feedback
A This is the complement. Use the Third Angles Theorem.B This is the supplement. Use the Third Angles Theorem.C The Third Angles Theorem states that if two angles of one triangle are congruent to two
angles of another triangle, then the third pair of angles are congruent.D Correct!
PTS: 1 DIF: Advanced NAT: 12.3.3.f STA: GE12.0TOP: 4-2 Angle Relationships in Triangles
15. ANS: A
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem, ∠ACB ≅ ∠DCE. Therefore, ∆CBA ≅ ∆CED by ASA.
Feedback
A Correct!B Adjacent angles are angles in a plane that have their vertex and one side in common but
have no interior points in common. Angle ACB and angle DCE are not adjacent angles.C Use ASA, not SAS, to prove the triangles congruent.D Look for vertical angles.
PTS: 1 DIF: Basic REF: Page 253 OBJ: 4-5.2 Applying ASA CongruenceNAT: 12.3.2.e STA: GE5.0 TOP: 4-5 Triangle Congruence: ASA AAS and HL
ID: A
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16. ANS: AThe image of (x,y) is (x − 1, y + 3).A(3, 4)→ A(3− 1, 4+ 3) = A′(1, 7)
B(2, − 1) → B(2− 1, − 1+ 3) = B′(1, 2)
C(4, 12)→ C(4− 1, 12+ 3) = C ′(3, 15)
Feedback
A Correct!B Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.C Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.D Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.
PTS: 1 DIF: Average REF: Page 833 OBJ: 12-2.3 Drawing Translations in the Coordinate Plane NAT: 12.3.2.cSTA: GE22.0 TOP: 12-2 Translations
17. ANS: C[1] Steps 1 and 7 state that two angles and a nonincluded side of ∆MLN and ∆PLO are congruent. By AAS, ∆MLN ≅ ∆PLO.
[2] Since ∆MLN ≅ ∆PLO, by CPCTC, ML ≅ PL.
Feedback
A Steps 1 and 7 state that two angles and a nonincluded side of triangle MLN and triangle PLO are congruent. Which triangle congruence theorem states that the triangles are congruent?
B Before using CPCTC, you must prove that triangle MLN and triangle PLO are congruent.
C Correct!D Before using CPCTC, you must prove that triangle MLN and triangle PLO are
congruent. Since steps 1 and 7 state that two angles and a nonincluded side are congruent, which triangle congruence theorem states that the triangles are congruent?
PTS: 1 DIF: Average REF: Page 261 OBJ: 4-6.3 Using CPCTC in a ProofNAT: 12.3.5.a STA: GE5.0 TOP: 4-6 Triangle Congruence: CPCTC
ID: A
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18. ANS: BSubstitute 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 The angles are alternate interior angles.B Correct!C The lanes are parallel by the Converse of the Alternate Interior Angles Theorem.D The angles are alternate interior angles.
PTS: 1 DIF: Average REF: Page 165 OBJ: 3-3.4 ApplicationNAT: 12.3.5.a STA: GE7.0 TOP: 3-3 Proving Lines Parallel
19. ANS: D∆ABC is equilateral. Equiangular triangles are equilateral.2s − 10 = s + 2 Definition of equilateral triangle.s = 12 Subtract s and add 10 to both sides of the equation.
AB = 2s − 10AB = 2 12( ) − 10 Substitute 12 for s in the equation for AB.AB = 14 Simplify.
CA = AB Definition of equilateral triangle.CA = 14 Substitute 14 for AB.
Feedback
A Equiangular triangles are equilateral. Use AB = BC to solve for s, and then use AC = AB or AC = BC to find AC.
B By a corollary to the Isosceles Triangle Theorem, equiangular triangles are equilateral. Use AB = BC to solve for s, and then use AC = AB or AC = BC to find AC.
C This is s. Substitute s in the original equation to find AC.D Correct!
PTS: 1 DIF: Basic REF: Page 275 OBJ: 4-8.3 Using Properties of Equilateral Triangles NAT: 12.3.3.fSTA: GE12.0 TOP: 4-8 Isosceles and Equilateral Triangles
ID: A
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20. ANS: AProof:
Statements Reasons1. ∠1 and ∠2 are supplementary. 1. Given2. m∠1 = 135° 2. Given3. m∠1 + m∠2 = 180° 3. Definition of supplementary angles4. 135° + m∠2 = 180° 4. Substitution Property5. m∠2 = 45° 5. Subtraction Property of Equality
Feedback
A Correct!B To get from Step 4 to Step 5, use subtraction, not substitution.C The angles are supplementary, not complementary.D Check to the given information.
PTS: 1 DIF: Average REF: Page 111 OBJ: 2-6.2 Completing a Two-Column Proof NAT: 12.3.5.aSTA: GE2.0 TOP: 2-6 Geometric Proof
21. ANS: CEG = FH Given informationEG = EF + FG Segment Addition PostulateFH = FG + GH Segment Addition PostulateEF + FG = FG + GH Substitution Property of EqualityEF = GH Subtraction Property of Equality
Feedback
A Check the properties.B Check the steps.C Correct!D The Angle Addition Postulate refers to angles, not segments.
PTS: 1 DIF: Average REF: Page 110 OBJ: 2-6.1 Writing JustificationsNAT: 12.3.5.a STA: GE2.0 TOP: 2-6 Geometric Proof
ID: A
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22. ANS: AProof:
Statements Reasons
1. RT ⊥SU 1. Given
2. ∠RTS and ∠RTU are right angles. 2. Definition of perpendicular lines3. ∠RTS ≅ ∠RTU 3. Right Angle Congruence Theorem4. ∠SRT ≅ ∠URT 4. Given5. ∠S ≅ ∠U 5. Third Angles Theorem
6. RS ≅ RU 6. Given
7. T is the midpoint of SU . 7. Given
8. ST ≅ UT 8. Definition of midpoint
9. RT ≅ RT 9. Reflexive Property of Congruence
10. ∆RTS ≅ ∆RTU 10. Definition of congruent triangles
Feedback
A Correct!B Use the correct property to show that the part is congruent to itself.C Use the definition of perpendicular lines to show that the lines intersect to form right
angles.D Angle S and angle U are not vertical angles. Use a different justification for Reason 5.
PTS: 1 DIF: Average REF: Page 232 OBJ: 4-3.3 Proving Triangles CongruentNAT: 12.3.5.a STA: GE5.0 TOP: 4-3 Congruent Triangles
23. ANS: BA reflection is a transformation that moves a figure (the preimage) by flipping it across a line.
Feedback
A See if you can flip the image across the line to get a congruent image.B Correct!
PTS: 1 DIF: Basic REF: Page 824 OBJ: 12-1.1 Identifying ReflectionsNAT: 12.3.2.c STA: GE22.0 TOP: 12-1 Reflections
ID: A
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24. ANS: D(3x)° = (4x − 24)° Alternate Exterior Angles Theorem−x = −24 Subtract 4x from both sides.x = 24 Divide both sides by −1.
m∠RST = 3x = 3(24)= 72° Substitute 24 for x .
Feedback
A After finding x, substitute to find the angle measure.B Find the measure of angle RST, not the value of x.C Find the measure of angle RST, not the supplement.D Correct!
PTS: 1 DIF: Average REF: Page 156 OBJ: 3-2.2 Finding Angle MeasuresNAT: 12.3.3.g STA: GE7.0 TOP: 3-2 Angles Formed by Parallel Lines and Transversals
25. ANS: DStep 1 Draw line l parallel to lines m and n. Step 2 Find m∠x. m∠1 = m∠x + m∠y Use the Corresponding Angles Postulate with
lines m and l. m∠x = 35°.
Step 3 Find m∠y. Use the Same-Side Interior Angles Theorem with lines l and n. m∠y = 180− 130= 50°.
Step 4 Find m∠1.m∠1 = m∠x + m∠y = 35+ 50= 85°
Feedback
A Use the Corresponding Angles Postulate and a theorem related to parallel lines and angle pairs.
B Use the Corresponding Angles Postulate and a theorem related to parallel lines and angle pairs.
C Use the Corresponding Angles Postulate and a theorem related to parallel lines and angle pairs.
D Correct!
PTS: 1 DIF: Advanced NAT: 12.2.1.f STA: GE7.0TOP: 3-2 Angles Formed by Parallel Lines and Transversals KEY: multi-step
ID: A
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26. ANS: DSince ∠1 ≅ ∠3, m∠1 ≅ m∠3.Thus m∠3 = 26.5°.
Since ∠3 and ∠4 are complementary, m∠4 = 90° − 26.5° = 63.5°.
Since ∠4 ≅ ∠6, m∠4 ≅ m∠6.Thus m∠6 = 63.5°.
By the Angle Addition Postulate,180° = m∠4+ m∠5+ m∠6
= 63.5° + m∠5+ 63.5°Thus, m∠5 = 53°.
Feedback
A Angle 1 and angle 3 are congruent. Congruent angles have the same measure.B Angle 3 and angle 4 are complementary, not supplementary.C The measure of angle 5 is 180 degrees minus the sum of the measure of angle 4 and the
measure of angle 6.D Correct!
PTS: 1 DIF: Average REF: Page 30 OBJ: 1-4.4 Problem-Solving ApplicationNAT: 12.3.3.g STA: 6MG2.2 TOP: 1-4 Pairs of Angles KEY: application | complementary angles | supplementary angles
27. ANS: CIn a flowchart, reasons follow statements. Using the two-column proof, the statement that leads to Reason 2 is ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary. The reason that follows Statement 3 is Congruent Supplements Theorem.
Feedback
A In a flowchart, reasons follow statements.B Angles 1 and 2 are supplements, not complements.C Correct!D In a flowchart, reasons follow statements.
PTS: 1 DIF: Average REF: Page 119 OBJ: 2-7.2 Writing a Flowchart ProofNAT: 12.3.5.a STA: GE2.0 TOP: 2-7 Flowchart and Paragraph Proofs
ID: A
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28. ANS: DStep 1 Draw line l parallel to lines m and n.Given: m∠y + m∠z = 90°, ∠x ≅ ∠w, m Ä n Ä l
Step 2 Use the Alternate Interior Angles Theorem to find pairs of congruent angles.
∠y ≅ ∠x, ∠z ≅ ∠w
m∠y = m∠x, m∠z = m∠wStep 3 Substitute x for y and w for z in the given m∠y + m∠z = 90°.
m∠x + m∠w = 90°Step 4 Use the definition of congruent angles and the given ∠x ≅ ∠w.
m∠x = m∠wStep 5 To find m∠w, substitute w for x.
m∠x + m∠w = 90°m∠w + m∠w = 90°
2 ⋅ m∠w = 90°m∠w = 45°
Step 6 Find m∠1. ∠1 and ∠w are supplementary.
m∠1+ m∠w = 180°m∠1+ 45° = 180°
m∠1 = 135°
Feedback
A Draw a line parallel to the given parallel lines and use the Alternate Interior Angles Theorem.
B Draw a line parallel to the given parallel lines and use the Alternate Interior Angles Theorem.
C Draw a line parallel to the given parallel lines and use the Alternate Interior Angles Theorem.
D Correct!
PTS: 1 DIF: Advanced NAT: 12.2.1.f STA: GE7.0TOP: 3-4 Perpendicular Lines KEY: multi-step
ID: A
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29. ANS: A
Rotation 90° about the origin of any point A(x, y) results in the image A′(−y, x). To rotate the point about
M(−7, − 9), determine hä, the horizontal vector, and vä, the vertical vector, from G to M. Then move
hävertically from M, and move the opposite of vä horizontally from M. The result of the rotation is labeled G ′ in the graph.
The line y = −6 is a horizontal line passing through (0, − 6). Reflection across a horizontal line involves movement of the point to the other side of the line, such that the image is the same distance from the line that the original point was. The x-coordinate does not change. The result of the reflection is labeled G″ in the graph.
Feedback
A Correct!B The point is reflected across a horizontal line, not a vertical line.C Rotation occurs before reflection.D Rotate around the given point, not around the origin.
PTS: 1 DIF: Advanced NAT: 12.3.2.c STA: GE22.0TOP: 12-4 Compositions of Transformations
ID: A
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30. ANS: BProof:
Statements Reasons
1. P is the midpoint of TQ and RS . 1. Given
2. TP ≅ QP, RP ≅ SP 2. Definition of midpoint
3. ∠TPR ≅ ∠QPS 3. Vertical Angles Theorem4. ∆TPR ≅ ∆QPS 4. SAS
Feedback
A There is not enough information to show that segment RT is congruent to segment SQ.B Correct!C Use the correct postulate to prove the triangles congruent.D Angle PRT and angle PSQ are not vertical angles.
PTS: 1 DIF: Average REF: Page 244 OBJ: 4-4.4 Proving Triangles CongruentNAT: 12.3.5.a STA: GE5.0 TOP: 4-4 Triangle Congruence: SSS and SAS
31. ANS: D
In the diagram, rays FG→
and FH→
share a common endpoint F and form the line GH→←
.
Feedback
A Opposite rays form a line.B Opposite rays are two rays that have a common endpoint and form a line.C Opposite rays form a line.D Correct!
PTS: 1 DIF: Basic REF: Page 7 OBJ: 1-1.2 Drawing Segments and RaysNAT: 12.3.1.d STA: GE1.0 TOP: 1-1 Understanding Points Lines and PlanesKEY: opposite rays
ID: A
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32. ANS: C
Statements 2 and 3 determine the measures of two angles of the triangle.Statement 1 determines the length of the included side.By ASA, the triangle must be unique.
Feedback
A There is not enough information for SAS. Draw a diagram to help you.B Draw a diagram. There is enough information to determine a unique triangle.C Correct!D There is not enough information for SSS. Draw a diagram to help you.
PTS: 1 DIF: Average REF: Page 252 OBJ: 4-5.1 Problem-Solving ApplicationNAT: 12.3.3.f STA: 7MR3.1 TOP: 4-5 Triangle Congruence: ASA AAS and HL
33. ANS: BA decagon has 10 sides and 10 vertices.sum of exterior angle measures = 360° Polygon Exterior Angle Sum Theorem
measure of one exterior angle = 36010
= 36°A regular decagon has 10 congruent exterior angles, so divide the sum by 10.
The measure of each exterior angle of a regular decagon is 36°.
Feedback
A Divide 360 by the number of sides.B Correct!C Divide 360 by the number of sides the polygon has.D Divide by the number of sides the polygon has.
PTS: 1 DIF: Average REF: Page 384 OBJ: 6-1.4 Finding Exterior Angle Measures in Polygons NAT: 12.3.3.fSTA: GE12.0 TOP: 6-1 Properties and Attributes of Polygons
ID: A
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34. ANS: AA translation is a transformation where all the points of a figure are moved the same distance in the same direction.
This transformation is a translation because all of the points have moved the same distance in the same direction.
Feedback
A Correct!B Check where all the points have moved.
PTS: 1 DIF: Basic REF: Page 831 OBJ: 12-2.1 Identifying TranslationsNAT: 12.3.2.c STA: GE22.0 TOP: 12-2 Translations
35. ANS: DCorresponding 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 Corresponding angles lie on the same side of a transversal, on the same sides of two lines.
B Angle 4 and angle 1 are supplementary angles, not corresponding angles.C Angle 5 and angle 7 are vertical angles, not corresponding angles.D Correct!
PTS: 1 DIF: Basic REF: Page 147 OBJ: 3-1.2 Classifying Pairs of AnglesNAT: 12.3.3.g STA: GE7.0 TOP: 3-1 Lines and Angles
ID: A
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36. ANS: B
: This appears to be a reflection.
A rotation is a transformation that turns a figure around a fixed point, called the center of rotation. If the transformation is a rotation, then the figure on the left rotates clockwise 90° about a fixed point to look like this:
Feedback
A See if you can rotate the image around a fixed point and get a congruent image.B Correct!
PTS: 1 DIF: Basic REF: Page 839 OBJ: 12-3.1 Identifying RotationsNAT: 12.3.2.c STA: GE22.0 TOP: 12-3 Rotations
37. ANS: A1a. By the Linear Pair Theorem, ∠CBF and ∠ABC are supplementary and ∠CDG and ∠ADC are supplementary. 1b. Given ∠CBF ≅ ∠CDG, by the Congruent Supplements Theorem, ∠ABC ≅ ∠ADC.
2. ∠CAB ≅ ∠CAD by the definition of an angle bisector.
3. AC ≅ AC by the Reflexive Property of Congruence
4. Two angles and a nonincluded side of ∆ACB and ∆ACD are congruent. By AAS, ∆ACB ≅ ∆ACD.
5. Since ∆ACB ≅ ∆ACD, AD ≅ AB by CPCTC.
Feedback
A Correct!B For reason 1, check whether the linear pairs are complementary or supplementary.C For statement 2, use the fact that line segment AC bisects angle A, not angle C.D Find the correct property that states that a line segment is congruent to itself.
PTS: 1 DIF: Average REF: Page 260 OBJ: 4-6.2 Proving Corresponding Parts Congruent NAT: 12.3.5.aSTA: GE5.0 TOP: 4-6 Triangle Congruence: CPCTC
ID: A
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38. ANS: A
1. ∠B and ∠G are alternate interior angles and AB Ä GH . Thus by the Alternate Interior Angles Theorem, ∠B ≅ ∠G.
2. ∠ACB and ∠HFG are alternate exterior angles and AC→←
Ä FH→←
. Thus by the Alternate Exterior Angles Theorem, ∠ACB ≅ ∠HFG.
Feedback
A Correct!B If line AC is parallel to line FG, are angle ACB and angle HFG alternate interior angles
or alternate exterior angles?C You switched the definitions of alternate interior and alternate exterior angles.D If line segment AB is parallel to line segment GH, are angle B and angle G alternate
exterior angles or alternate interior angles?
PTS: 1 DIF: Average REF: Page 254 OBJ: 4-5.3 Using AAS to Prove Triangles Congruent NAT: 12.3.5.aSTA: GE5.0 TOP: 4-5 Triangle Congruence: ASA AAS and HL
39. ANS: CThe image of (x, y) is (–y, x).R(4, –1) → R′(1, 4)S(5, 3) → S ′(–3, 5)Q(3, 1) → Q ′(–1, 3)Graph the preimage and the image.
Feedback
A This is a rotation by 180° about the origin.B This is a reflection across the y-axis.C Correct!D The rotation is 90° counterclockwise, not clockwise.
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ID: A
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40. ANS: CStep 1 Find the sum of the interior angle measures.(n – 2)180° Polygon Angle Sum Theorem= (45 – 2)180° A 45-gon has 45 sides, so substitute 45 for n.= 7740 Simplify.
Step 2 Find the measure of one interior angle.774045
= 172 The interior angles are ≅, so divide by 45.
Feedback
A Subtract, not add, 2 from the number of sides.B Subtract 2, not 1, from the number of sides.C Correct!D According to the Polygon Angle Sum Theorem, the sum of the interior angle measures
is the product of 180 and the number of sides minus 2.
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41. ANS: BProof:
Statements Reasons1. t ⊥ l,∠1 ≅ ∠2 1. Given2. t ⊥ m 2. If 2 intersecting lines form linear pair of ≅
∠s → lines ⊥.3. m Ä l 3. If 2 lines ⊥ to the same line → lines Ä.
Feedback
A Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the lines are parallel.
B Correct!C Switch Reason 2 and Reason 3.D Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines
are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the lines are parallel.
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ID: A
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42. ANS: DIn a flowchart, reasons flow from the statement above. The statement above Reason 2 is AB + BF = FC + CD. The statement above Reason 3 is AB + BF = AF ; FC + CD = FD.
Feedback
A Reasons flow from the statement above.B Reasons flow from the statement above.C Reasons flow from the statement above.D Correct!
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43. ANS: BTo 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 To find which line is the transversal for a given angle pair, locate the line that connects the vertices.
B Correct!C Alternate interior angles lie on opposite sides of the transversal, between two lines.D To find which line is the transversal for a given angle pair, locate the line that connects
the vertices.
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44. ANS: AAB = a + 7 = 3+ 7 = 10CD = 4a − 2 = 4(3)− 2 = 12− 2 = 10AD = 6a − 2 = 6(3)− 2 = 18− 2 = 16CB = 16
AB ≅ CD. AD ≅ CB. BD ≅ BD by the Reflexive Property of Congruence. So ∆ABD ≅ ∆CDB by SSS.
Feedback
A Correct!B Substitute 3 for a.C Check the measures of segment AD and segment CB.D Use the correct postulate.
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ID: A
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45. ANS: C
∠FAC and ∠3 are adjacent angles. Their noncommon sides, AF→
and AG→
, are opposite rays, so ∠FAC and ∠3 also form a linear pair.
Feedback
A Adjacent angles form a linear pair if and only if their noncommon sides are opposite rays.
B Two angles are adjacent if they have a common vertex and a common side, but no common interior points.
C Correct!
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46. ANS: D∠E ≅ ∠N Third Angles Theoremm∠E = m∠N Definition of congruent angles
(x2)° = (4x2 − 75)° Substitute x2 for m∠E and 4x2 − 75 for m∠N .
−3x2 = −75 Subtract 4x2 from both sides.
x2 = 25 Divide both sides by –3.
So m∠E = 25°.Since m∠E = m∠N , m∠N = 25°.
Feedback
A These are the measures of angles F and P, not angles E and N.B Use the Third Angles Theorem.C The Third Angles Theorem states that if two angles of one triangle are congruent to two
angles of another triangle, then the third pair of angles are congruent.D Correct!
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ID: A
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47. ANS: BCE = CD + DE Segment Addition Postulate6x = 4x + 8( ) + 27 Substitute 6x for CE and 4x + 8 for CD.6x = 4x + 35 Simplify.2x = 35 Subtract 4x from both sides.2x2
= 352
Divide both sides by 2.
x = 352
or 17.5 Simplify.
CE = 6x = 6 17.5( ) = 105
Feedback
A You found the value of x. Find the length of the specified segment.B Correct!C Check your equation. Make sure you are not subtracting instead of adding.D You found the length of a different segment.
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48. ANS: BA translation is a transformation where all the points of a figure are moved the same distance in the same direction.
This transformation is not a translation because not all of the points have moved the same distance.
Feedback
A Check where all the points have moved.B Correct!
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49. ANS: D(x)° = (3x − 70)° Corresponding Angles Postulate0 = 2x − 70 Subtract x from both sides.70 = 2x Add 70 to both sides.35 = x Divide both sides by 2.
m∠ABC = 3x − 70m∠ABC = 3(35)− 70= 35° Substitute 35 for x. Simplify.
Feedback
A Use the Corresponding Angles Postulate.B First, set the measures of the corresponding angles equal to each other. Then, solve for
x and substitute in the expression (3x – 70).C If two parallel lines are cut by a transversal, then the pairs of corresponding angles are
congruent.D Correct!
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50. ANS: BProof:
Statements Reasons1. m∠1 + m∠2 = 90° 1. Given2. m∠3 + m∠4 = 90° 2. Given3. m∠1 + m∠2 = m∠3 + m∠4 3. Substitution Property4. m∠2 = m∠3 4. Given5. m∠1 + m∠2 = m∠2 + m∠4 5. Substitution Property6. m∠1 = m∠4 6. Subtraction Property of Equality
Feedback
A To get from Step 4 to Step 5, use substitution, not subtraction.B Correct!C To get from Step 4 to Step 5, use substitution, not addition.D Check the given information.
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