Geometry Workbook — Chapter 2: Parallel Lines Cut by a Transversal

SOL G.RLT.2 (a, b, c) · Companion to Textbook Chapter 2

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 106.

PAGE 1 — Chapter opener

Chapter 2 · Parallel Lines Cut by a Transversal

Standard G.RLT.2 (a, b, c)

In this chapter you will:

• Name the eight angles a transversal makes and sort them into interior and exterior
• Identify the five named pairs and say whether each is congruent or supplementary
• Write a two-column proof of each relationship from the Corresponding Angles Postulate
• Find every angle from one given measure, naming the relationship at each step
• Solve for a variable when measures are algebraic, choosing the right equation
• Use the converses to prove two lines are parallel — or that they are not

Words to know: transversal · interior · exterior · corresponding · alternate interior · alternate exterior · same-side interior · same-side exterior · vertical angles · linear pair · two-column proof · converse

Convention: alternate means congruent, same-side means supplementary. Angles are numbered 1–4 at line $m$ (above, then below) and 5–8 the same way at line $n$.

PAGE 2 — The eight angles

2.1 The Transversal

FIGURE: fig1-transversal-and-eight-angles.png (full width)

Fill in the blanks.

Interior angles lie ____________________ the two parallel lines: ∠, ∠, ∠, ∠

Exterior angles lie ____________________ them: ∠, ∠, ∠, ∠

1. List the interior angles and the exterior angles.

   Interior: ____________ Exterior: ____________

2. Which angle is the upper-left angle at the lower intersection? ______

3. Name the four interior angles. ____________

PAGE 3 — The five named pairs

The Five Pairs

FIGURE: fig2-five-named-angle-pairs.png (full width)

Complete the frame.

3. Name the pair type in each panel and say whether it is congruent or supplementary.

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4. State the pattern in one sentence.

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PAGE 4 — Naming pairs

Practice · Name the Pair

4. ∠1 and ∠5: ____________________

5. ∠3 and ∠6: ____________________

6. ∠4 and ∠6: ____________________ What does that mean about their measures? ____________

7. Classify each pair.

   a) ∠2 and ∠6 ____________________

   b) ∠4 and ∠5 ____________________

   c) ∠1 and ∠8 ____________________

   d) ∠3 and ∠5 ____________________

   e) ∠2 and ∠8 ____________________

8. For each pair in item 7, congruent or supplementary?

   a) ______ b) ______ c) ______ d) ______ e) ______

PAGE 5 — Practice and apply · Lesson 2.1

Practice

FIGURE: fig12-blank-transversal-frames.png (left two thirds)

9. Both alternate interior pairs: ____________ Both same-side interior pairs: ____________

10. Both alternate exterior pairs: ____________ Both same-side exterior pairs: ____________

11. $m\angle 1 = 105°$. Find $m\angle 5$ ______, $m\angle 4$ ______, $m\angle 8$ ______, naming each relationship.

12. Every angle congruent to ∠7: ____________

13. Draw your own transversal figure in the blank frame and mark one pair of each of the five types.

14. ∠1 and ∠7: ____________________ Congruent or supplementary? ______

15. $m\angle 6 = 74°$. $m\angle 4$ = ______

PAGE 6 — Reasoning · Lesson 2.1

Think It Through

13. Application. Two parallel rows of desks are cut by a diagonal aisle. The aisle makes a $58°$ angle with the first row on the upper-right. What angle does it make with the second row in the matching position? Which relationship says so?

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14. Reasoning. Why can same-side interior angles not also be congruent unless both measure $90°$?

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15. Error analysis. A student says ∠3 and ∠5 are alternate interior because both are interior. What is the error? Name the pair correctly.

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16. If the transversal is perpendicular to both lines, all eight angles measure ______ . Which relationships still hold? ____________

17. Reasoning. Two lines are cut by a transversal and ∠2 ≇ ∠6. What does that tell you about the lines?

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PAGE 7 — The postulate everything rests on

2.2 Proving the Relationships

FIGURE: fig3-corresponding-angles-postulate.png (full width)

Fill in the blanks.

Corresponding Angles Postulate. If two parallel lines are cut by a transversal, then each pair of corresponding angles is ____________________ .

This statement is ____________________ (assumed / proved). Everything else in the chapter is ____________________ from it.

25. State the Corresponding Angles Postulate in your own words.

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26. Which relationship is assumed as a postulate? ____________________

PAGE 8 — Anatomy of a two-column proof

What a Proof Looks Like

FIGURE: fig6-two-column-proof-anatomy.png (full width)

23. What is the "Given"? ____________________ What is being proved? ____________________

24. What reason justifies statement 4? ____________________

Three rules.

• The first reason is almost always ____________________ .
• The last statement is exactly ____________________ .
• No line may use a fact that ____________________ .

43. Why is "the picture shows it" not a valid reason?

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PAGE 9 — Alternate interior angles

Alternate Interior Angles

FIGURE: fig4-alternate-interior-angles-proof.png (full width)

26. Which two facts combine to give ∠3 ≅ ∠6?

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27. Give the two reasons that combine to prove the Alternate Interior Angles Theorem.

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28. Complete the proof that ∠4 ≅ ∠5, given $m \parallel n$.

PAGE 10 — Same-side interior angles

Why Same-Side Is Different

FIGURE: fig5-same-side-interior-supplementary.png (full width)

27. Add the two marked measures. Sum = ______ What does that tell you? ____________________

28. Why does the same-side interior proof switch from ≅ to = partway through?

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29. Where does the "supplementary" in the same-side theorems come from? ____________________

30. Prove that ∠3 and ∠5 are supplementary, given $m \parallel n$.

PAGE 11 — More proofs

Practice · Proofs

FIGURE: fig12-blank-transversal-frames.png (right third — blank proof frame)

30. Prove ∠1 ≅ ∠8 (alternate exterior), given $m \parallel n$.

31. Prove ∠2 and ∠8 are supplementary, given $m \parallel n$.

32. Give the reason for each step.

    a) ∠1 ≅ ∠4 ____________________

    b) ∠1 ≅ ∠5 ____________________

    c) $m\angle 1 + m\angle 3 = 180°$ ____________________

33. Write a two-column proof of your own for any one of the five relationships, using angles other than the ones above.

PAGE 12 — Reasoning · Lesson 2.2

Think It Through

34. Error analysis. A proof's second line reads "∠3 ≅ ∠6, Reason: Corresponding Angles Postulate." What is the error? Correct the reason.

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35. Error analysis. A proof concludes "∠4 ≅ ∠6" from the Same-Side Interior Angles Theorem. What went wrong?

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36. Reasoning. Why must one of the five relationships be assumed rather than proved?

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37. Reasoning. Rewrite the alternate interior proof as a paragraph proof.

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38. Application. A carpenter measures the angle a brace makes with the first of two parallel rafters as $47°$ and predicts $47°$ at the second rafter. Write the two-step justification.

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39. Why may "vertical angles are congruent" be used here without being proved?

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PAGE 13 — A numeric angle chase

2.3 Angle Chases

FIGURE: fig7-numeric-angle-chase.png (full width)

45. $m\angle 1$ = ______

46. $m\angle 2$ = ______ Relationship: ____________________

47. $m\angle 5$ = ______ Relationship: ____________________

48. $m\angle 6$ = ______ Relationship: ____________________

Fill in the blanks. Only ______ distinct measures ever appear in this figure, and they are ____________________ .

PAGE 14 — Practice · numeric chases

Practice · Find Them All

51. $m\angle 4 = 128°$. Fill in every angle and name a relationship for each.

52. $m\angle 7 = 39°$. $m\angle 1$ = ______ $m\angle 4$ = ______ $m\angle 6$ = ______

53. $m\angle 2 = 81°$. $m\angle 6$ = ______ Relationship: ____________________

PAGE 15 — Algebraic measures

Which Equation?

FIGURE: fig8-algebraic-angle-measures.png (full width)

49. Left panel — pair type: ____________________ Equation: ____________________

50. Right panel — pair type: ____________________ Equation: ____________________

The one decision.

Congruent pair → set the two expressions ____________________ .

Same-side pair → set their sum to ____________________ .

66. State that decision in one sentence.

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PAGE 16 — Practice · solving for a variable

Practice · Solve

53. Corresponding angles $(5x + 8)°$ and $(3x + 28)°$. $x$ = ______ Measure = ______ Check: ______

54. Alternate interior $(6x - 14)°$ and $(4x + 6)°$. $x$ = ______ Measure = ______

55. Same-side interior $(3x + 12)°$ and $(2x - 2)°$. $x$ = ______ Measures = ______ and ______

56. Same-side exterior $(4x + 15)°$ and $(x + 10)°$. $x$ = ______ Measures = ______ and ______

57. Alternate exterior $(9x - 40)°$ and $(5x + 8)°$. $x$ = ______ Then find every other angle.

58. Alternate interior $(2x + 9)°$ and $(x + 25)°$. $x$ = ______

59. Same-side interior $(4x)°$ and $(2x + 60)°$. $x$ = ______

PAGE 17 — Apply and reason · Lesson 2.3

Apply It

58. Application. A ramp runs between two parallel handrails, making a $23°$ angle with the lower rail on the upper-right. Angle with the upper rail in the same position: ______ Angle on the other side of the ramp at the upper rail: ______

59. Application. Two parallel garden beds are crossed by a path with same-side interior angles $(2x + 5)°$ and $(3x)°$. $x$ = ______ Angles = ______ and ______

60. Error analysis. Given alternate interior angles $(x + 40)°$ and $(3x)°$, a student writes $x + 40 + 3x = 180$. Name the error, solve correctly, give both measures.

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61. Reasoning. Why do only two distinct measures ever appear, and what do they add to?

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62. Reasoning. If solving gives an angle measure of $190°$, what do you know before checking your arithmetic?

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PAGE 18 — Running the theorems backwards

2.4 Proving Lines Are Parallel

FIGURE: fig9-proving-lines-are-parallel.png (full width)

Fill in the blanks.

Each of the five theorems has a ____________________ , and all five are ____________________ .

Given congruent corresponding angles → cite the ____________________ .

67. Which pair is marked, and what are their measures? ____________________

68. Which converse does the figure let you cite? ____________________

69. In the two-column proof, what reason justifies step 2? ____________________

70. State the converse of the Same-Side Exterior Angles Theorem.

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PAGE 19 — Proving lines are NOT parallel

When the Test Refuses

FIGURE: fig10-not-parallel-counterexample.png (full width)

70. What do the two marked angles sum to? ______

71. Why does that sum show the lines are not parallel?

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72. Must the lines be parallel? Name the converse, or say why none applies.

    a) ∠1 ≅ ∠5 ____________________

    b) ∠3 ≅ ∠6 ____________________

    c) $m\angle 4 = 100°$, $m\angle 6 = 80°$ ____________________

    d) $m\angle 3 = 70°$, $m\angle 5 = 70°$ ____________________

    e) ∠2 ≅ ∠3 ____________________

PAGE 20 — Proofs and solving · Lesson 2.4

Practice

74. Given ∠1 ≅ ∠8, prove $m \parallel n$.

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

76. Corresponding $(5x - 12)°$ and $(3x + 8)°$. Parallel when $x$ = ______

77. Alternate exterior $(7x + 4)°$ and $(9x - 20)°$. Parallel when $x$ = ______

78. Same-side interior $(6x + 10)°$ and $(4x)°$. Parallel when $x$ = ______

79. Corresponding $(3x)°$ and $(x + 50)°$. Parallel when $x$ = ______

PAGE 21 — Apply and reason · Lesson 2.4

Apply It

79. Application. A straightedge crosses two grout lines at $63°$ and $63°$ in corresponding position. Is the work square? Cite the converse.

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80. Application. Another straightedge gives same-side interior angles of $88°$ and $94°$. Parallel? What would the second angle have to be? ____________

81. Error analysis. Given ∠2 ≅ ∠3, a student concludes $m \parallel n$. Why does that not follow?

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82. Reasoning. Difference between what 2.2 proves and what 2.4 proves, in terms of hypothesis and conclusion:

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83. Reasoning. All five converses here are true. Why does Chapter 1's warning still matter?

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84. Draw a figure whose marked angles prove two lines are not parallel, and justify in one sentence.

85. ∠3 ≅ ∠6. Parallel? ______ Converse cited: ____________________

86. $m\angle 4 = 95°$, $m\angle 6 = 85°$. Parallel? ______ Justify: ____________________

87. How does a theorem from 2.2 differ from its converse in 2.4?

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PAGE 22 — Context

2.5 Parallel Lines in Context

FIGURE: fig11-parallel-lines-in-context.png (full width)

Two jobs before you name a pair.

1. Identify which objects are the ____________________ , and say what makes them so.

2. Identify the ____________________ , which is whatever crosses both.

89. Parallel pair: ____________________ Transversal: ____________________

90. Measure marked at the south curb: ______

91. Measure marked "?": ______ Relationship: ____________________

92. Angle between the crosswalk and the north curb on the other side: ______

93. If the crosswalk met the south curb at $70°$, the corresponding angle at the north curb would be ______

94. One other real situation with parallel lines and a transversal: ____________________

PAGE 23 — Context practice

Apply It

95. Application. A walkway meets the near of two parallel rails at $52°$. Corresponding angle at the far rail: ______ Angle on the other side of the walkway at the far rail: ______

96. Application. A ladder makes a $71°$ angle with the lower of two parallel rungs. Same-side interior angle at the upper rung: ______

97. Application. A curb makes $105°$ and $75°$ in the same-side interior position with two parking rows. Parallel? ______ Justify: ____________________

98. Application. Another curb gives same-side interior angles of $98°$ and $86°$. Parallel? ______ Off by how many degrees? ______

99. Application. Alternate interior angles $(3x + 6)°$ and $(x + 34)°$ on two rafters. $x$ = ______ Measure = ______

100. Application. Same-side exterior $(2x + 10)°$ and $(3x)°$ on two lane lines. $x$ = ______ Measures = ______ and ______

PAGE 24 — Reasoning and exit · Lesson 2.5

Check Yourself

101. Reasoning. A problem says "the shelves are level." Does that make them parallel? What are you assuming?

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102. Error analysis. A student reports the corresponding angle at the north curb as $116°$ instead of $64°$. What is the error?

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103. Reasoning. How do you tell whether a context problem wants a theorem or a converse?

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104. Write your own context problem: two parallel lines, a transversal, one given angle, two asked for.

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105. What two things must you identify before naming an angle pair in context? ____________________

106. Two parallel curbs are crossed by a path at $48°$. Give all eight angle measures and name two relationships.

     ∠1 ______ ∠2 ______ ∠3 ______ ∠4 ______ ∠5 ______ ∠6 ______ ∠7 ______ ∠8 ______

PAGE 25 — Chapter review

Review

Review 1 (G.RLT.2a). $m \parallel n$ cut by $t$, with $m\angle 1 = 117°$.

• All eight measures: ∠1 ______ ∠2 ______ ∠3 ______ ∠4 ______ ∠5 ______ ∠6 ______ ∠7 ______ ∠8 ______
• ∠1 with ∠5: ____________ ______ · ∠1 with ∠7: ____________ ______ · ∠1 with ∠8: ____________ ______ · ∠3 with ∠5: ____________ ______ · ∠3 with ∠6: ____________ ______
• Two-column proof that ∠4 ≅ ∠5:

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

• Parallel when $x$ = ______ Measures = ______ and ______
• Two-column proof that the lines are parallel at that value:
• If $x = 15$: measures ______ and ______ · parallel? ______ · justify: ____________

Review 3 (G.RLT.2c). Two parallel handrails crossed by a support meeting the lower rail at $57°$ uphill.

• Corresponding angle at the upper rail: ______

• Same-side interior angle at the upper rail: ______

• A second support gives $57°$ and $59°$ in the corresponding position. What does that tell you? What would have to change?

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Canva production notes

• Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
• Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
• Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white.
• Proof frames on pages 9, 10, 11, and 20 are typed two-column tables with ruled blanks, not figures. The blank five-row proof frame in fig12-blank-transversal-frames.png is for item 40 and any extra proof a teacher assigns.
• Blank transversal diagrams in fig12 are unlabeled on purpose — students write the angle numbers themselves on page 5 (item 18) and use them for the extra practice on page 20.
• Symbols: ∠, ≅, ≇, ∥, and the degree sign must render in the body font; check the Canva font supports them before the first export.
• Item numbers are continuous from 1 to 106 and must not be renumbered when pages are reordered.