Geometry Workbook — Chapter 5: Proving Triangles Congruent

SOL G.TR.2 (a, d, e) · Companion to Textbook Chapter 5

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 126.

PAGE 1 — Chapter opener

Chapter 5 · Proving Triangles Congruent

Standard G.TR.2 (a, d, e)

In this chapter you will:

• Write a correspondence in order, and use CPCTC to unpack a congruence into six facts
• Prove triangles congruent by SSS and SAS, and say what makes an angle included
• Prove triangles congruent by ASA and AAS
• Use HL on right triangles, and explain why SSA and AAA fail
• Write both a direct and an indirect proof
• Construct a congruent triangle with compass and straightedge
• Find measured attributes of a triangle you cannot reach

Words to know: congruent · correspondence · CPCTC · included angle · included side · SSS · SAS · ASA · AAS · HL · SSA · AAA · Reflexive Property · vertical angles · indirect proof · construction

Convention: order carries the claim. $\triangle ABC \cong \triangle DEF$ asserts $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$. CPCTC comes last, never first.

PAGE 2 — Correspondence

5.1 Corresponding Parts

FIGURE: fig1-correspondence-in-order.png (full width)

Fill in the blanks.

Congruent triangles have ______ pairs of congruent parts: ______ pairs of sides and ______ pairs of angles.

The ____________________ of the letters in a congruence statement is part of the claim.

1. Which vertex of $\triangle DEF$ corresponds to $B$? ______

2. Which side corresponds to $\overline{AC}$? ______

3. Write the six congruences asserted by $\triangle ABC \cong \triangle DEF$.

   ---

   ---

4. Why is $\triangle ABC \cong \triangle EFD$ a different claim?

   ---

PAGE 3 — CPCTC

The Reason That Unpacks It

FIGURE: fig2-cpctc-unpacked.png (full width)

Fill in the blanks.

CPCTC stands for ____________________________________________ .

It may be used only ____________ the triangles have been proved congruent.

4. What must be true before CPCTC may be used? ____________________

5. State CPCTC in your own words. ____________________________

6. When may CPCTC be used? ____________________

PAGE 4 — Practice · corresponding parts

Practice

7. $\triangle MNO \cong \triangle PQR$. a) $\overline{MN} \cong$ ______ b) $\angle O \cong$ ______ c) $\overline{PR} \cong$ ______

8. $\triangle ABC \cong \triangle DEF$, $AB = 7$, $BC = 9$, $AC = 5$. $DE$ ______ $EF$ ______ $DF$ ______

9. $\triangle STU \cong \triangle VWX$, $m\angle S = 40°$, $m\angle T = 75°$. $m\angle V$ ______ $m\angle W$ ______ $m\angle X$ ______

10. $\triangle ABC \cong \triangle DEF$. Angle congruent to $\angle E$: ______ Side congruent to $\overline{EF}$: ______

11. $J \leftrightarrow R$, $K \leftrightarrow S$, $L \leftrightarrow T$. Congruence statement: ____________________

12. A student writes $\triangle JKL \cong \triangle STR$. Which correspondences are wrong? ____________________

13. $\triangle ABC$ and $\triangle ADC$ share $\overline{AC}$. What corresponds to $\overline{AC}$, and what does that tell you?

    ---

14. $\triangle ABC \cong \triangle DEF$. Angle corresponding to $\angle C$: ______

15. $\triangle PQR \cong \triangle XYZ$, $QR = 14$. $YZ$ = ______ Reason: ____________

PAGE 5 — Apply and reason · Lesson 5.1

Think It Through

14. Application. Two identical tabletops from one template; one edge measures $31$ in. What is known about the second, and why?

    ---

15. Application. Truss triangles $\triangle ABC \cong \triangle DEF$, $m\angle B = 58°$. $m\angle E$ = ______ Reason: ____________

16. Error analysis. "$\triangle ABC \cong \triangle DEF$, so $\overline{AB} \cong \overline{EF}$." Error and correction:

    ---

17. Reasoning. Why does congruence give six facts rather than three?

    ---

18. Reasoning. Why may CPCTC never appear before the congruence statement?

    ---

19. Why is the order of the letters part of the claim? ____________________

PAGE 6 — SSS and SAS

5.2 SSS and SAS

FIGURE: fig3-sss-and-sas.png (full width)

Complete the criteria frame.

23. Left panel: ____________________ Criterion: ______

24. Right panel: ____________________ Criterion: ______

25. State SAS using the word "included." ____________________________

PAGE 7 — The included angle

Where Is the Angle?

FIGURE: fig4-included-angle.png (full width)

Fill in the blanks.

An angle is included between two sides when its ____________ is an endpoint of ____________ .

Mark the two ____________ first, then look at where the angle is.

25. Which angle is included between $\overline{AB}$ and $\overline{AC}$? ______

26. Why is $\angle B$ not included between them?

    ---

27. Two sides and an angle not between them is called ______ . Is it a criterion? ______

PAGE 8 — Practice · SSS and SAS

Practice

29. Criterion, or "none."

    a) $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\overline{CA} \cong \overline{FD}$ ______

    b) $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$, $\overline{BC} \cong \overline{EF}$ ______

    c) $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\angle A \cong \angle D$ ______

30. In $\triangle XYZ$, the angle included between $\overline{XY}$ and $\overline{YZ}$: ______

31. In $\triangle XYZ$, the two sides that include $\angle Z$: ______

32. $\triangle ABC$ and $\triangle CDA$ share $\overline{AC}$. Write the congruence: ____________ Reason: ____________

33. Given $\overline{JK} \cong \overline{MN}$, $\overline{KL} \cong \overline{NP}$. One more for SAS: ____________ One more for SSS: ____________

34. Three pairs of congruent sides is ______

35. In $\triangle ABC$, the angle included between $\overline{BA}$ and $\overline{BC}$: ______

36. $\overline{PQ} \cong \overline{XY}$, $\overline{QR} \cong \overline{YZ}$, $\angle Q \cong \angle Y$. Criterion: ______

37. Two triangles share a side. Congruence: ____________ Reason: ____________

PAGE 9 — Proofs · Lesson 5.2

Write the Proof

FIGURE: fig14-blank-congruence-frames.png (middle third — blank proof)

34. Given $\overline{AB} \cong \overline{CB}$ and $\overline{AD} \cong \overline{CD}$, prove $\triangle ABD \cong \triangle CBD$.

35. Given $\overline{RS} \cong \overline{RU}$ and $\angle SRT \cong \angle URT$, prove $\triangle RST \cong \triangle RUT$.

PAGE 10 — Apply and reason · Lesson 5.2

Apply It

36. Application. A gate braced by a diagonal splits a rectangular frame into two triangles. Criterion: ______ Why?

    ---

37. Application. Two tent panels each with edges $6$, $6$, $4$ ft. Criterion: ______

38. Error analysis. Given $\overline{AB} \cong \overline{DE}$, $\overline{AC} \cong \overline{DF}$, $\angle C \cong \angle F$, a student writes "SAS." Error:

    ---

39. Reasoning. Why does SSS work? Use what Chapter 4 established.

    ---

40. Reasoning. Two triangles share a side and have one more congruent pair. Fewest extra givens for SSS: ______ For SAS: ______

PAGE 11 — ASA and AAS

5.3 ASA and AAS

FIGURE: fig5-asa-and-aas.png (full width)

45. Left panel marks: ____________________

46. Right panel marks: ____________________ Difference: ____________

47. The side included between $\angle A$ and $\angle B$ in $\triangle ABC$: ______

48. Why is AAS a valid criterion?

    ---

49. Why does knowing two pairs of angles give the third? ____________________

PAGE 12 — The free givens

Two Givens Nobody Writes Down

FIGURE: fig9-shared-side-and-vertical-angles.png (full width)

Fill in the blanks.

A shared side is congruent to itself by the ____________________ .

Two segments that cross give a congruent pair of ____________________ .

49. What congruence does the shared diagonal give, and by what reason? ____________________

50. What congruence does the crossing give, and by what reason? ____________________

51. Two segments cross. Free congruence: ____________ Reason: ____________

PAGE 13 — Practice · ASA and AAS

Practice

51. Criterion, or "none."

    a) $\angle A \cong \angle D$, $\overline{AC} \cong \overline{DF}$, $\angle C \cong \angle F$ ______

    b) $\angle A \cong \angle D$, $\angle C \cong \angle F$, $\overline{AB} \cong \overline{DE}$ ______

    c) $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\angle C \cong \angle F$ ______

52. In $\triangle PQR$, the side included between $\angle P$ and $\angle R$: ______

53. In $\triangle PQR$, the two angles that include $\overline{QR}$: ______

54. Given $\angle X \cong \angle A$, $\angle Y \cong \angle B$. One more for ASA: ____________ One more for AAS: ____________

55. Which criterion does item 56 use? ______ Which free given? ____________

56. Why does $\angle C \cong \angle F$ follow from two pairs of congruent angles?

    ---

57. Two angles and the side between them is ______

58. $\angle P \cong \angle S$, $\angle R \cong \angle U$, $\overline{PQ} \cong \overline{ST}$. ASA or AAS? ______

PAGE 14 — Proofs · Lesson 5.3

Write the Proof

55. Given $\overline{AC}$ and $\overline{BD}$ intersect at $E$, $\overline{AE} \cong \overline{CE}$, $\angle BAE \cong \angle DCE$; prove $\triangle ABE \cong \triangle CDE$.

56. Given $\angle B \cong \angle D$, $\angle BAC \cong \angle DAC$; prove $\triangle ABC \cong \triangle ADC$.

PAGE 15 — Apply and reason · Lesson 5.3

Apply It

59. Application. A surveyor's two triangles share a segment and have vertical angles at a crossing; one angle at each bank is $47°$. Criterion: ______ Free given used: ____________

60. Application. Two gables with equal $38°$ base angles and equal $9$-ft ridge beams between them. Criterion: ______

61. Error analysis. Given $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\overline{AC} \cong \overline{DF}$, a student writes "ASA." Error:

    ---

62. Reasoning. Why is there no SSA criterion but there is an AAS one?

    ---

PAGE 16 — HL

5.4 HL, for Right Triangles Only

FIGURE: fig6-hl-right-triangles.png (full width)

Fill in the blanks.

HL applies only to ____________ triangles. It is the only criterion that names a ____________________ .

67. What three things are marked in each triangle? ____________________

68. Why can HL not be applied without a right angle?

    ---

69. State HL, including its restriction. ____________________________

70. Why is HL safe even though its parts form an SSA arrangement?

    ---

PAGE 17 — The two that fail

SSA and AAA

FIGURE: fig7-ssa-ambiguous-case.png (full width)

69. What are the two triangles, and which parts do they share? ____________________

70. Why does the arc from $B$ meet the ray twice?

    ---

FIGURE: fig8-aaa-is-not-congruence.png (full width)

71. What is congruent, and what is not? ____________________

72. What relationship does AAA establish? ____________ Which chapter? ______

73. The two combinations that are not criteria: ____________ and ____________

74. Why does AAA fail? ____________________

Fill in the blank. Every valid criterion contains at least one ______ .

PAGE 18 — Practice · Lesson 5.4

Practice

73. Must they be congruent? Name the criterion or the reason none applies.

    a) two right triangles, congruent hypotenuses, one pair of congruent legs ____________

    b) $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\angle C \cong \angle F$ ____________

    c) all three pairs of angles congruent ____________

    d) $\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$ ____________

74. Two right triangles, both pairs of legs congruent. Criterion: ______ Is HL needed? ______

75. In the SSA figure, what if the swung side were $8$?

    ---

76. What if it were exactly the perpendicular distance from $B$ to the ray?

    ---

77. HL proof: given $\angle B$, $\angle E$ right angles, $\overline{AC} \cong \overline{DF}$, $\overline{AB} \cong \overline{DE}$; prove $\triangle ABC \cong \triangle DEF$.

PAGE 19 — Apply and reason · Lesson 5.4

Apply It

78. Application. Two ramps with equal sloped edges and equal post heights. Criterion: ______ What must be true of the posts? ____________

79. Application. Two sails, all angles matching, one a scale model. Congruent? ______ What are they? ____________

80. Error analysis. A student proves congruence by "SSA." What is wrong, and what is the correct conclusion?

    ---

81. Error analysis. A student uses HL with no right angles marked. Error:

    ---

82. Reasoning. Why does every valid criterion contain at least one S?

    ---

PAGE 20 — The direct proof

5.5 Direct and Indirect Proofs

FIGURE: fig10-two-column-congruence-proof.png (full width)

Complete the proof-shape frame.

Write the ____________ ; add the ____________ congruence; name the ____________ ; use ____________ for the part asked for.

87. Given: ____________________ Prove: ____________

88. Which step is the free given? ______ Reason: ____________________

89. Which step names the criterion? ______ Which criterion? ______

90. Which step uses CPCTC? ______ Why can it not come earlier?

    ---

91. Which step must come immediately before CPCTC? ____________________

92. What reason justifies a shared side? ____________________

PAGE 21 — The indirect proof

Assume the Opposite

FIGURE: fig11-indirect-proof-structure.png (full width)

Fill in the blanks.

The first line must state the ____________________ , the negation of the conclusion.

The last line must name the ____________________ and reject the assumption.

91. What must the first line do? ____________________

92. What must the last line do? ____________________

93. First line of an indirect proof of each.

    a) $\triangle ABC \cong \triangle DEF$ ____________________

    b) $m\angle A > 60°$ ____________________

    c) two lines are parallel ____________________

94. First line of an indirect proof of "$\triangle PQR \cong \triangle STU$": ____________________

95. Name the two things an indirect proof must do. ____________________

PAGE 22 — Practice · proofs

Write the Proofs

93. Given $\overline{AB} \cong \overline{CB}$, $\overline{AD} \cong \overline{CD}$; prove $\angle A \cong \angle C$.

94. Given $\overline{PQ} \parallel \overline{SR}$ and $\overline{PQ} \cong \overline{SR}$; prove $\triangle PQR \cong \triangle RSP$.

95. Reasons: a) $\overline{XY} \cong \overline{XY}$ ____________ b) vertical $\angle 1 \cong \angle 2$ ____________ c) $\overline{AB} \cong \overline{DE}$ right after a congruence ____________

96. Rewrite item 93 as a paragraph proof.

    ---

PAGE 23 — Indirect practice and apply

Apply It

98. Prove indirectly that a triangle cannot have two obtuse angles.

    ---

99. Given $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, $\angle B \cong \angle E$; prove $\triangle ABC \cong \triangle DEF$ indirectly.

    ---

100. Application. A bridge inspector has both pairs of outer edges congruent and the included angles equal. Write her two-column proof.

101. Error analysis. Step 2 is "CPCTC" and step 5 is the congruence. Error:

_______________________________________________

102. Reasoning. Why has an indirect proof with no contradiction proved nothing?

_______________________________________________

PAGE 24 — Constructing a congruent triangle

5.6 Construct a Copy

FIGURE: fig12-construct-a-congruent-triangle.png (full width)

Complete the construction steps.

(1) Copy one ____________ . (2) Swing an ____________ from each endpoint with the other two side lengths. (3) The arcs cross at the third ____________ . (4) Draw the remaining sides and ____________ the arcs.

107. What is drawn first in the copy? ____________________

108. What do the two arcs locate, and why that point? ____________________

109. Which criterion makes the copy congruent? ______

110. Why must the arcs be left on the page?

_______________________________________________

123. First step of the SSS construction: ____________________

124. Why are the arcs kept? ____________________

FIGURE: fig14-blank-congruence-frames.png (right third — construction frame)

PAGE 25 — Measured attributes

5.6 Reading Off What You Cannot Measure

FIGURE: fig13-measured-attributes-in-context.png (full width)

Complete the four-step method.

(1) Establish the ____________________ . (2) Write the ____________________ in order. (3) Apply ____________ . (4) Answer in ____________ with units.

111. Which criterion establishes the congruence? ______

112. Which reason gives the unknown side? ______

113. State the four-step method. ____________________________

PAGE 26 — Practice · Lesson 5.6

Practice

113. List the SSS construction steps in order. ____________________________

114. Describe the SAS construction instead. ____________________________

115. $\triangle ABC \cong \triangle DEF$, $AB = 8$, $BC = 13$, $CA = 10$. $DE$ ______ $EF$ ______ $FD$ ______

116. $\triangle JKL \cong \triangle MNP$, $m\angle J = 38°$, $m\angle K = 96°$. $m\angle M$ ______ $m\angle N$ ______ $m\angle P$ ______

117. $\triangle ABC \cong \triangle DEF$, $AC = 17$ m, $m\angle B = 72°$. $DF$ ______ $m\angle E$ ______ Reason: ______

118. $\triangle RST \cong \triangle XYZ$, $RS = 6$, $ST = 9$, $RT = 12$. A student says $XY = 12$. Error and correct value: ____________

119. $\triangle ABC \cong \triangle DEF$, $BC = 21$. $EF$ = ______ Reason: ____________

PAGE 27 — Apply and reason · Lesson 5.6

Apply It

119. Application. Two congruent beds; one has sides $4$, $6$, $7$ m and apex $80°$. Give the other's sides and apex. ____________________

120. Application. A surveyor's walkable triangle has sides $40$, $55$, $30$ m. Longest distance across the ravine: ______ Reason: ____________

121. Error analysis. $\triangle ABC \cong \triangle DEF$, $m\angle C = 41°$; a student writes $m\angle E = 41°$. Error:

     ---

122. Reasoning. Why does writing the correspondence first prevent the most common error?

     ---

PAGE 28 — Chapter review

Review

Review 1 (G.TR.2a). $\overline{AC}$ and $\overline{BD}$ meet at $E$; $\overline{AE} \cong \overline{CE}$, $\overline{BE} \cong \overline{DE}$.

• Free given from the crossing: ____________ Reason: ____________
• Two-column proof that $\triangle ABE \cong \triangle CDE$, with criterion:
• CPCTC conclusion: ____________ Why not earlier? ____________
• If $\overline{BE} \cong \overline{DE}$ became $\overline{AB} \cong \overline{CD}$: arrangement ______ , why it fails ____________ ; a replacement that works ____________ , criterion ______

Review 2 (G.TR.2a). Criterion or why none:

(i) three pairs of sides ______ (ii) two sides, non-included angle ______ (iii) two angles, non-included side ______ (iv) three angles ______ (v) right triangles, hypotenuses and one leg ______

Why (ii) fails but (v) succeeds: ____________________________

Review 3 (G.TR.2 d, e). Template: sides $9$, $12$, $15$ in; the angle opposite the $15$-in side is $90°$.

• SSS construction in order, and what must be left on the page: ____________________
• A bracket side of $12$ in corresponds to ______ ; other two sides ______ and ______
• Two criteria that could prove a bracket congruent, given the right angle: ______ and ______
• How CPCTC gets the angle opposite the $9$-in side from one measurement, and why measuring every bracket is unnecessary: ____________________

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, 14, 18, and 22 are typed two-column tables with ruled blanks, not figures. fig14-blank-congruence-frames.png supplies an unmarked congruent pair, a six-row proof frame, and a construction frame with one side already laid off — use them on pages 9, 24, and for any extra proof a teacher assigns.
• Constructions on pages 24 and 26 need room for compass work: leave at least 4 in of clear width, and instruct students not to erase arcs.
• Symbols: ≅, ≇, ∠, △, ∥, and the overline for segments must render in the body font; check the Canva font supports them before the first export.
• Item numbers are continuous from 1 to 126 and must not be renumbered when pages are reordered.