Appendix A — Answer Key, Chapter 2: Parallel Lines Cut by a Transversal

SOL G.RLT.2 (a, b, c) · Covers textbook Chapter 2 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 106 across the chapter. Proofs show one acceptable order of steps, not the only one.

Conventions used in every answer below: angles are numbered $1$–$4$ at line $m$ (above it, then below it, left before right) and $5$–$8$ the same way at line $n$; interior means between the lines; segments and angles are congruent while their measures are equal; every proof line carries a claim and a reason.

The five relationships, for reference:

Vertical pairs are $1$&$4$, $2$&$3$, $5$&$8$, $6$&$7$. In every such figure only two angle measures appear, and they are supplementary.

Lesson 2.1 — The Transversal and Its Eight Angles

Guided practice

1. Interior: $\angle 3$, $\angle 4$, $\angle 5$, $\angle 6$. Exterior: $\angle 1$, $\angle 2$, $\angle 7$, $\angle 8$.
2. $\angle 5$.
3. Corresponding (congruent); alternate interior (congruent); alternate exterior (congruent); same-side interior (supplementary); same-side exterior (supplementary).
4. Corresponding angles — congruent.
5. Alternate interior angles — congruent.
6. Same-side interior angles — supplementary, so their measures add to $180°$.

Independent practice

7. a) corresponding b) alternate interior c) alternate exterior d) same-side interior e) same-side exterior
8. a) congruent b) congruent c) congruent d) supplementary e) supplementary
9. Alternate interior: $\angle 3$ & $\angle 6$, and $\angle 4$ & $\angle 5$. Same-side interior: $\angle 3$ & $\angle 5$, and $\angle 4$ & $\angle 6$.
10. Alternate exterior: $\angle 1$ & $\angle 8$, and $\angle 2$ & $\angle 7$. Same-side exterior: $\angle 1$ & $\angle 7$, and $\angle 2$ & $\angle 8$.
11. $m\angle 5 = 105°$ (corresponding to $\angle 1$). $m\angle 4 = 105°$ (vertical to $\angle 1$). $m\angle 8 = 105°$ (alternate exterior to $\angle 1$; or corresponding to $\angle 4$).
12. $\angle 2$, $\angle 3$, and $\angle 6$.
13. $58°$, by the Corresponding Angles Postulate.
14. Same-side interior angles are supplementary, so their measures add to $180°$. If they were also congruent, each would be half of $180°$, which is $90°$. So the only way a same-side pair can be congruent is if the transversal is perpendicular to both lines.
15. Being interior is only half the test — the pair must also be on opposite sides of the transversal to be "alternate." $\angle 3$ and $\angle 5$ are both on the same side, so they are same-side interior, and therefore supplementary rather than congruent.
16. All eight measure $90°$. Every relationship still holds: the three congruent pairs are all $90° = 90°$, and the two supplementary pairs are $90° + 90° = 180°$. This is the one case where a same-side pair is also congruent.
17. The lines are not parallel. Corresponding angles are congruent whenever the lines are parallel, so a corresponding pair that is not congruent rules parallelism out.
18. Figures vary. A correct drawing has two lines marked parallel, a transversal crossing both, angles numbered $1$–$8$ in the standard positions, and one marked example of each of the five types.

Exit ticket 2.1

19. $\angle 3$, $\angle 4$, $\angle 5$, $\angle 6$.
20. Same-side exterior — supplementary.
21. $\angle 4$ and $\angle 6$ are same-side interior, so $m\angle 4 = 180° - 74° = 106°$.
22. Alternate pairs and corresponding pairs are congruent; same-side pairs are supplementary.

Lesson 2.2 — Proving the Angle Pair Relationships

Guided practice

23. Given: $m \parallel n$, cut by transversal $t$. Prove: $\angle 3 \cong \angle 6$.
24. Transitive Property of Congruence.
25. Sample: when a transversal crosses two parallel lines, any two angles sitting in matching positions at the two crossings have the same measure.
26. $\angle 2 \cong \angle 6$ (corresponding angles) and $\angle 2 \cong \angle 3$ (vertical angles). Transitivity joins them.
27. $122° + 58° = 180°$ — the two angles are supplementary.
28. Congruence is a relation between figures, and you cannot add congruences. The proof needs to add two measures to reach $180°$, so it converts $\cong$ to $=$ by the definition of congruent angles, does the arithmetic, and converts back at the end by the definition of supplementary angles.

Independent practice

29. Given $m \parallel n$, prove $\angle 4 \cong \angle 5$.

30. Given $m \parallel n$, prove $\angle 1 \cong \angle 8$.

31. Given $m \parallel n$, prove $\angle 3$ and $\angle 5$ are supplementary.

32. Given $m \parallel n$, prove $\angle 2$ and $\angle 8$ are supplementary.

33. a) Vertical angles are congruent. b) Corresponding Angles Postulate. c) Linear Pair Postulate.
34. $\angle 3$ and $\angle 6$ are alternate interior, not corresponding, so the postulate does not name them. Either cite the Alternate Interior Angles Theorem, or insert the two steps that prove it ($\angle 2 \cong \angle 6$ by the postulate, $\angle 2 \cong \angle 3$ by vertical angles, then transitivity).
35. The Same-Side Interior Angles Theorem concludes supplementary, not congruent. From it, $m\angle 4 + m\angle 6 = 180°$; congruence would additionally require both to be $90°$, which is not given.
36. A proof derives new statements from statements already accepted. If every statement had to be proved, there would be nothing to start from and the chain would never end. Geometry therefore assumes a small number of postulates; here the Corresponding Angles Postulate is the one assumed, and the other four relationships are theorems derived from it.
37. Sample paragraph proof: Because $m \parallel n$, the corresponding angles $\angle 2$ and $\angle 6$ are congruent by the Corresponding Angles Postulate. Also, $\angle 2$ and $\angle 3$ are vertical angles, so they are congruent. Since $\angle 3$ is congruent to $\angle 2$ and $\angle 2$ is congruent to $\angle 6$, the Transitive Property of Congruence gives $\angle 3 \cong \angle 6$. Therefore alternate interior angles are congruent.
38. The angle at the first rafter and the angle at the second rafter on the same side of the brace are corresponding, so they are congruent by the Corresponding Angles Postulate; the alternate interior angle at the second rafter is congruent to that corresponding angle because they are vertical angles. So the alternate interior angle is also $47°$.
39. It was proved in Grade 8 (8.MG.1) from the Linear Pair Postulate, and this chapter's objectives are the five parallel-line relationships. Reusing an already-proved result is exactly what a proof is allowed to do.
40. Proofs vary; a correct one starts from "Given: $m \parallel n$," cites the Corresponding Angles Postulate at least once, and reaches the stated relationship with a reason on every line.

Exit ticket 2.2

41. The Corresponding Angles Postulate.
42. The Corresponding Angles Postulate and "vertical angles are congruent," joined by the Transitive Property.
43. A figure may not be drawn to scale, and a proof must convince a reader who has not seen it. Appearance is not one of the accepted reasons — a definition, postulate, theorem, or property is.
44. From the Linear Pair Postulate. A same-side pair is a congruent pair with one angle swapped for that angle's linear-pair partner, and the linear pair contributes the $180°$.

Lesson 2.3 — Angle Chases, Numeric and Algebraic

Guided practice

45. $m\angle 1 = 106°$.
46. $m\angle 2 = 74°$ — linear pair with $\angle 1$.
47. $m\angle 5 = 106°$ — corresponding to $\angle 1$.
48. $m\angle 6 = 74°$ — corresponding to $\angle 2$ (also alternate interior to $\angle 3$).
49. Alternate interior, so the two expressions are equal: $3x + 10 = 5x - 30$.
50. Same-side interior, so the sum is $180°$: $(5y + 5) + (2y + 35) = 180$.

Independent practice

51. $m\angle 1 = 128°$ (vertical to $\angle 4$); $m\angle 2 = 52°$ (linear pair with $\angle 4$); $m\angle 3 = 52°$ (linear pair with $\angle 4$); $m\angle 5 = 128°$ (corresponding to $\angle 1$); $m\angle 6 = 52°$ (corresponding to $\angle 2$); $m\angle 7 = 52°$ (corresponding to $\angle 3$); $m\angle 8 = 128°$ (corresponding to $\angle 4$). Other routes are fine as long as a relationship is named.
52. $m\angle 1 = 141°$ — $\angle 1$ and $\angle 7$ are same-side exterior, so $180° - 39° = 141°$. $m\angle 4 = 141°$ — vertical to $\angle 1$. $m\angle 6 = 39°$ — vertical to $\angle 7$.
53. $5x + 8 = 3x + 28 \Rightarrow 2x = 20 \Rightarrow x = 10$. Measure $5(10) + 8 = 58°$; check $3(10) + 28 = 58°$.
54. $6x - 14 = 4x + 6 \Rightarrow 2x = 20 \Rightarrow x = 10$. Measure $6(10) - 14 = 46°$; check $4(10) + 6 = 46°$.
55. $(3x + 12) + (2x - 2) = 180 \Rightarrow 5x + 10 = 180 \Rightarrow x = 34$. Measures $114°$ and $66°$; check $114 + 66 = 180$.
56. $(4x + 15) + (x + 10) = 180 \Rightarrow 5x + 25 = 180 \Rightarrow x = 31$. Measures $139°$ and $41°$; check $139 + 41 = 180$.
57. $9x - 40 = 5x + 8 \Rightarrow 4x = 48 \Rightarrow x = 12$. Each measures $9(12) - 40 = 68°$. The figure then has $68°$ at $\angle 1$, $\angle 4$, $\angle 5$, $\angle 8$ and $112°$ at $\angle 2$, $\angle 3$, $\angle 6$, $\angle 7$ — the standard two-value pattern. (Which set is which depends on the figure's orientation; the point is that exactly two supplementary values appear.)
58. Corresponding angle at the upper rail: $23°$. On the other side of the ramp at the upper rail: $180° - 23° = 157°$ (linear pair).
59. $(2x + 5) + 3x = 180 \Rightarrow 5x + 5 = 180 \Rightarrow x = 35$. Angles $75°$ and $105°$.
60. Alternate interior angles are congruent, so the equation should be $x + 40 = 3x$, not a sum of $180$. Solving: $2x = 40$, $x = 20$; both measure $60°$.
61. The transversal cuts each parallel line into the same two supplementary wedges, and the parallelism makes the pattern at the second crossing identical to the pattern at the first. So every angle is one of those two values, and the two always add to $180°$ because each pair of adjacent angles is a linear pair.
62. An angle in this figure can never exceed $180°$, because every angle is one of two supplementary values. A measure of $190°$ means an error — most often the wrong equation (setting a congruent pair's sum to $180$, or a same-side pair equal) rather than the arithmetic.

Exit ticket 2.3

63. $\angle 2$ and $\angle 6$ are corresponding, so $m\angle 6 = 81°$.
64. $2x + 9 = x + 25 \Rightarrow x = 16$ (each angle measures $41°$).
65. $4x + (2x + 60) = 180 \Rightarrow 6x = 120 \Rightarrow x = 20$ (the angles are $80°$ and $100°$).
66. Name the pair: a congruent pair (corresponding, alternate interior, alternate exterior) means set the expressions equal; a same-side pair means set their sum to $180°$.

Lesson 2.4 — Proving Lines Are Parallel

Guided practice

67. Alternate interior angles $\angle 3$ and $\angle 6$, both $66°$.
68. The Converse of the Alternate Interior Angles Theorem.
69. Vertical angles are congruent.
70. $72° + 95° = 167°$.
71. If the lines were parallel, those same-side interior angles would have to be supplementary — they would sum to $180°$. They sum to $167°$, so the assumption fails and the lines are not parallel.
72. If two lines are cut by a transversal and a pair of same-side exterior angles is supplementary, then the lines are parallel.

Independent practice

73. a) Yes — Converse of the Corresponding Angles Postulate. b) Yes — Converse of the Alternate Interior Angles Theorem. c) Yes — $\angle 4$ and $\angle 6$ are same-side interior and $100 + 80 = 180$, so the Converse of the Same-Side Interior Angles Theorem applies. d) No conclusion. $\angle 3$ and $\angle 5$ are same-side interior, so parallelism would need them supplementary; two $70°$ angles sum to $140°$, so the lines are not parallel. e) No conclusion. $\angle 2$ and $\angle 3$ are vertical angles, so they are congruent whether or not the lines are parallel. A vertical pair carries no information about parallelism.
74. Given $\angle 1 \cong \angle 8$, prove $m \parallel n$.

75. Given $\angle 4$ and $\angle 6$ supplementary, prove $m \parallel n$.

76. $5x - 12 = 3x + 8 \Rightarrow 2x = 20 \Rightarrow x = 10$ (each angle $38°$).
77. $7x + 4 = 9x - 20 \Rightarrow 2x = 24 \Rightarrow x = 12$ (each angle $88°$).
78. $(6x + 10) + 4x = 180 \Rightarrow 10x = 170 \Rightarrow x = 17$ (angles $112°$ and $68°$).
79. Yes, the work is square. The two $63°$ angles are corresponding, so the Converse of the Corresponding Angles Postulate gives parallel grout lines.
80. No. Same-side interior angles must be supplementary, and $88 + 94 = 182 \ne 180$. For the lines to be parallel the second angle would have to be $180° - 88° = 92°$ — the work is off by $2°$.
81. $\angle 2$ and $\angle 3$ are vertical angles. Vertical angles are congruent in every figure, parallel or not, so their congruence is not evidence of anything. None of the five converses applies to a vertical pair.
82. Lesson 2.2 takes "the lines are parallel" as the hypothesis and an angle relationship as the conclusion. Lesson 2.4 swaps them: an angle relationship is the hypothesis and "the lines are parallel" is the conclusion. They are converses of each other.
83. Because a converse being true has to be established, not assumed. Chapter 1's point was that the truth of $p \rightarrow q$ tells you nothing about $q \rightarrow p$; here all five converses happen to be true, but each is a separate theorem that had to be proved. If students take "the converse came free" as the lesson, they will apply it in Chapter 5 or 10, where a converse is often false.
84. Figures vary. A correct one marks a pair whose relationship fails — for example alternate interior angles of $70°$ and $75°$, or same-side interior angles of $80°$ and $85°$ — with a sentence such as "alternate interior angles would have to be congruent, and $70 \ne 75$, so the lines are not parallel."

Exit ticket 2.4

85. Yes. Alternate interior angles are congruent, so the Converse of the Alternate Interior Angles Theorem gives $m \parallel n$.
86. Yes. $\angle 4$ and $\angle 6$ are same-side interior and $95 + 85 = 180$, so they are supplementary and the converse applies.
87. $3x = x + 50 \Rightarrow 2x = 50 \Rightarrow x = 25$ (each angle $75°$).
88. A theorem in 2.2 assumes parallel lines and concludes an angle relationship; its converse in 2.4 assumes the angle relationship and concludes the lines are parallel.

Lesson 2.5 — Parallel Lines in Context

Guided practice

89. Parallel lines: the north and south curbs. Transversal: the crosswalk.
90. $64°$.
91. $64°$, by the Corresponding Angles Postulate.
92. $180° - 64° = 116°$ (linear pair with the corresponding angle; equivalently a same-side interior pair with the $64°$ angle at the south curb).
93. $70°$.
94. Samples: rails of a train track crossed by a tie or a maintenance walkway; courses of bricks crossed by a diagonal expansion joint; rows of a crop field crossed by an irrigation line; parallel rafters crossed by a collar tie; ruled lines on paper crossed by a diagonal fold.

Independent practice

95. Corresponding angle at the far rail: $52°$. On the other side of the walkway at the far rail: $180° - 52° = 128°$.
96. $180° - 71° = 109°$.
97. Yes. $105° + 75° = 180°$, so the same-side interior angles are supplementary and the Converse of the Same-Side Interior Angles Theorem applies.
98. No. $98 + 86 = 184 \ne 180$. For parallel rows the second angle would have to be $82°$, so the row is off by $4°$.
99. $3x + 6 = x + 34 \Rightarrow 2x = 28 \Rightarrow x = 14$. Each angle measures $3(14) + 6 = 48°$; check $14 + 34 = 48°$.
100. $(2x + 10) + 3x = 180 \Rightarrow 5x = 170 \Rightarrow x = 34$. Measures $78°$ and $102°$; check $78 + 102 = 180$.
101. Two level shelves are parallel if they lie in the same vertical plane — which is the assumption. "Level" means each is horizontal, and two horizontal lines in the same plane are parallel; two horizontal lines in different planes can be skew. In a textbook context the same-plane assumption is intended, but naming it is the honest answer.
102. The student reported the supplement of the correct angle. The corresponding angle is congruent to the given one, so it is $64°$; $116°$ is the angle on the other side of the crosswalk at the north curb, which is a different angle.
103. Look at what is given and what is asked. If the problem states the lines are parallel and asks for a measure, use a theorem. If the problem gives measures and asks whether the lines are parallel — "is it square?", "are the rows straight?" — use a converse.
104. Problems vary. A complete one names the two parallel objects, names the crossing object, gives one angle measure, and asks for two others that can be found by named relationships.

Exit ticket 2.5

105. Which objects are the parallel lines (and why they are parallel), and which object is the transversal.
106. The two values are $48°$ and $132°$. With the standard numbering and the path rising to the right: $m\angle 1 = 132°$, $m\angle 2 = 48°$, $m\angle 3 = 48°$, $m\angle 4 = 132°$, $m\angle 5 = 132°$, $m\angle 6 = 48°$, $m\angle 7 = 48°$, $m\angle 8 = 132°$. Sample relationships: $\angle 2$ and $\angle 6$ are corresponding, so both are $48°$; $\angle 4$ and $\angle 6$ are same-side interior, and $132 + 48 = 180$.

Chapter 2 Review

Review 1 (G.RLT.2a). With $m\angle 1 = 117°$:

• $m\angle 1 = 117°$, $m\angle 2 = 63°$, $m\angle 3 = 63°$, $m\angle 4 = 117°$, $m\angle 5 = 117°$, $m\angle 6 = 63°$, $m\angle 7 = 63°$, $m\angle 8 = 117°$.
• $\angle 1$ and $\angle 5$: corresponding, congruent ($117°$ and $117°$). $\angle 1$ and $\angle 7$: same-side exterior, supplementary ($117° + 63° = 180°$). $\angle 1$ and $\angle 8$: alternate exterior, congruent ($117°$ and $117°$). $\angle 3$ and $\angle 5$: same-side interior, supplementary ($63° + 117° = 180°$). $\angle 3$ and $\angle 6$: alternate interior, congruent ($63°$ and $63°$).
• Proof that $\angle 4 \cong \angle 5$:

Review 2 (G.RLT.2b). Same-side interior angles $(7x - 4)°$ and $(3x + 24)°$.

• Parallel requires supplementary: $(7x - 4) + (3x + 24) = 180 \Rightarrow 10x + 20 = 180 \Rightarrow x = 16$. Measures $7(16) - 4 = 108°$ and $3(16) + 24 = 72°$; check $108 + 72 = 180$.
• Proof at $x = 16$:

• At $x = 15$ the measures are $7(15) - 4 = 101°$ and $3(15) + 24 = 69°$. Their sum is $170°$, not $180°$, so the same-side interior angles are not supplementary and the lines are not parallel.

Review 3 (G.RLT.2c).

• Corresponding angle at the upper rail: $57°$ (Corresponding Angles Postulate).
• Same-side interior angle at the upper rail: $180° - 57° = 123°$.
• The second support gives corresponding angles of $57°$ and $59°$. Corresponding angles of parallel lines must be congruent, so the rails are not parallel — at least, not as that second support meets them. For the rails to be parallel the second angle would have to be $57°$, so the upper rail (or the support's placement) is off by $2°$. Note that the first support's readings were consistent with parallel rails; one consistent measurement does not prove parallelism, and one inconsistent measurement disproves it.

Workbook-only items

Page 2, fill in the blanks. Interior angles lie between the two parallel lines: $\angle 3$, $\angle 4$, $\angle 5$, $\angle 6$. Exterior angles lie outside them: $\angle 1$, $\angle 2$, $\angle 7$, $\angle 8$.

Page 3, pair frame. Corresponding $\angle 2$ & $\angle 6$, congruent. Alternate interior $\angle 3$ & $\angle 6$, congruent. Alternate exterior $\angle 2$ & $\angle 7$, congruent. Same-side interior $\angle 4$ & $\angle 6$, supplementary. Same-side exterior $\angle 1$ & $\angle 7$, supplementary.

Page 7, fill in the blanks. …then each pair of corresponding angles is congruent. This statement is assumed. Everything else is proved from it.

Page 8, three rules. The first reason is almost always Given. The last statement is exactly what you were asked to prove. No line may use a fact that has not already appeared above it.

Page 13, fill in the blanks. Only two distinct measures ever appear, and they are supplementary.

Page 15, the one decision. Congruent pair → set the two expressions equal. Same-side pair → set their sum to $180°$.

Page 18, fill in the blanks. Each of the five theorems has a converse, and all five are true. Given congruent corresponding angles → cite the Converse of the Corresponding Angles Postulate.

Page 22, two jobs. (1) Identify which objects are the parallel lines, and say what makes them parallel. (2) Identify the transversal, which is whatever crosses both.

Pages 5, 11, and 20, blank frames. Any assigned figure or proof. Expected conventions: parallel lines carry matching arrowheads; angles are numbered in the standard positions; every proof line has both a claim and a reason, the first reason is "Given," and the final statement is the one asked for.