Geometry Workbook — Chapter 7: Proving Triangles Similar

SOL G.TR.3 (a, b, c, d, e) · Companion to Textbook Chapter 7

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 144.

PAGE 1 — Chapter opener

Chapter 7 · Proving Triangles Similar

Standard G.TR.3 (a, b, c, d, e)

In this chapter you will:

• Read a similarity statement and give the scale factor in either direction
• Prove similarity by AA, SSS, and SAS
• Spot the configuration that gives AA most often — a line parallel to one side
• Turn a similarity statement into a proportion and solve it
• Prove similarity from coordinates: distance reaches SSS, slope reaches AA
• Describe a sequence of transformations — a dilation, then rigid motions
• Solve attribute problems, including measuring what you cannot reach

Words to know: similar · scale factor · proportional · proportion · cross products · AA · SSS similarity · SAS similarity · included angle · dilation · centre of dilation · rigid motion · similarity transformation · sequence · indirect measurement · perimeter

Conventions: the tilde means proportional, not equal. A scale factor has a direction — the new triangle's side goes on top. Scale factors are positive. A sequence is ordered.

PAGE 2 — Scale factor

7.1 Similar Means Proportional

FIGURE: fig1-similar-triangles-and-scale-factor.png (full width)

Fill in the blanks.

Similar triangles have congruent ____________ and proportional ____________ .

The constant ratio is the ____________________ , written $k$.

1. Scale factor from $\triangle ABC$ to $\triangle DEF$: ______

2. Side of $\triangle DEF$ corresponding to $\overline{AC}$: ______

3. Scale factor from $\triangle DEF$ to $\triangle ABC$: ______

PAGE 3 — Six claims

Three Ratios and Three Angles

FIGURE: fig2-correspondence-and-ratios.png (full width)

Fill in the blanks.

$\triangle ABC \sim \triangle DEF$ makes ______ claims: ______ ratios and ______ angle congruences.

Perimeter scales by ______ . What scaling does to area is Chapter ______ .

4. Show $\tfrac{DE}{AB}$ and $\tfrac{EF}{BC}$ reduce to the same value.

   ______ = ______ and ______ = ______

5. List all six claims $\triangle ABC \sim \triangle DEF$ makes.

   ---

6. Why does the order of the letters matter? ____________________

PAGE 4 — Practice · scale factor

Practice

7. $\triangle ABC \sim \triangle PQR$, $AB = 4$, $PQ = 10$. $k =$ ______

8. $\triangle ABC \sim \triangle DEF$, $k = 3$, $AB = 7$. $DE =$ ______

9. $\triangle JKL \sim \triangle MNP$, $JK = 12$, $MN = 8$, $KL = 15$. $NP =$ ______

10. $\triangle ABC \sim \triangle DEF$, $AB = 6$, $BC = 9$, $AC = 12$, $DE = 10$. $EF =$ ______ $DF =$ ______

11. $\triangle RST \sim \triangle XYZ$, $m\angle R = 43°$, $m\angle S = 76°$. $m\angle X$ ______ $m\angle Y$ ______ $m\angle Z$ ______

12. Sides $3$, $4$, $5$ and $9$, $12$, $15$. Similar? ______ $k =$ ______

13. Sides $4$, $6$, $8$ and $6$, $9$, $11$. Similar? ______ Check that settles it: ____________

14. $\triangle ABC \sim \triangle DEF$, $k = \tfrac{5}{4}$, perimeter of $\triangle ABC = 36$. Perimeter of $\triangle DEF =$ ______

15. $\triangle ABC \sim \triangle DEF$, $AB = 5$, $DE = 15$. $k =$ ______

16. $\triangle ABC \sim \triangle DEF$, $k = \tfrac{1}{2}$, $BC = 14$. $EF =$ ______

17. $\triangle ABC \sim \triangle DEF$, $m\angle A = 55°$, $m\angle E = 80°$. $m\angle C =$ ______

PAGE 5 — Apply and reason · Lesson 7.1

Think It Through

15. Error analysis. $AB = 8$, $DE = 12$; a student reports $k = \tfrac{2}{3}$ from $\triangle ABC$ to $\triangle DEF$. Correct them.

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16. Reasoning. Why are congruent triangles always similar, but similar triangles not always congruent?

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17. State in full what a similarity statement claims.

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PAGE 6 — AA

7.2 Two Angles Are Enough

FIGURE: fig3-aa-two-angles-are-enough.png (full width)

Fill in the blanks.

If two angles of one triangle are congruent to two angles of another, the triangles are ____________ .

The third pair is free because of the ____________________ .

AA fixes the ____________ but not the ____________ .

21. Which two angle measures are marked in both triangles? ______ and ______

22. Why is the third pair congruent without being marked? ____________________

23. Are the two triangles congruent? ______ Explain: ____________________

PAGE 7 — The parallel cut

A Line Parallel to One Side

FIGURE: fig4-parallel-cut-gives-similar-triangles.png (full width)

Fill in the blanks.

The ratios are a ____________________ , not an assumption.

24. Name the two angle congruences that give AA, with a reason for each.

    ____________________ Reason: ____________

    ____________________ Reason: ____________

25. $\tfrac{AD}{AB} =$ ______

26. $\tfrac{DE}{BC} =$ ______ How do you know without measuring? ____________________

PAGE 8 — Practice · AA

Practice

27. Angles $35°$, $80°$ and $80°$, $65°$. Similar? ______ Why: ____________

28. Angles $40°$, $60°$ and $40°$, $70°$. Similar? ______ Why: ____________

29. $\overline{DE} \parallel \overline{BC}$; $AD = 6$, $DB = 4$, $AE = 9$. $AB$ ______ $AC$ ______ $EC$ ______

30. $\overline{DE} \parallel \overline{BC}$; $AD = 8$, $AB = 20$, $DE = 6$. $BC =$ ______

31. $\overline{DE} \parallel \overline{BC}$; $AD = 5$, $DB = 7$, $AE = 10$. $AC$ ______ $EC$ ______

32. Two right triangles each contain a $32°$ angle. Similar? ______ The two pairs: ____________

33. AA with $m\angle A = m\angle D = 90°$, $m\angle B = m\angle E = 37°$. $m\angle C$ ______ $m\angle F$ ______

34. Angles $25°$, $105°$ and $105°$, $50°$. Similar? ______ Why: ____________

35. $\overline{DE} \parallel \overline{BC}$; $AD = 4$, $AB = 10$, $AE = 6$. $AC =$ ______

PAGE 9 — Apply and reason · Lesson 7.2

Think It Through

34. Application. A ramp rises $2$ ft over a $24$ ft run. A longer ramp of the same slope has a $36$ ft run.

    Why the triangles are similar: ____________________ $k =$ ______ Rise = ______ ft

35. Error analysis. A student says one pair of congruent angles makes two triangles similar. Counterexample:

    ---

36. Reasoning. Why does AA need only two pairs when similarity asks for three?

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37. Why is AA not a congruence criterion? ____________________

38. Name the two angle facts behind a parallel cut. ____________________

PAGE 10 — SSS similarity

7.3 Three Ratios

FIGURE: fig5-sss-similarity.png (full width)

Fill in the blanks.

SSS similarity asks for three ____________ , not three ____________ .

41. The three ratios: ______ , ______ , ______ Common value: ______

42. What would two matching ratios out of three prove? ____________________

PAGE 11 — SAS similarity

The Angle Must Be Included

FIGURE: fig6-sas-similarity.png (full width)

43. Which two sides and which angle does SAS similarity use? ____________________

44. Why must the congruent angle be the included one? ____________________

PAGE 12 — Congruence inside similarity

Congruence Is the Case k = 1

FIGURE: fig7-congruence-is-similarity-with-k-one.png (full width)

45. Which row changes between congruence and similarity? ______

46. Which arrangement fails for both? ______

PAGE 13 — Practice · SSS and SAS

Practice

47. Sides $6$, $9$, $12$ and $8$, $12$, $16$. Similar? ______ Criterion ______ $k =$ ______

48. Sides $5$, $7$, $9$ and $10$, $14$, $20$. Similar? ______ Comparison that settles it: ____________

49. Sides $4$, $6$, $8$ and $10$, $15$, $20$. Similar? ______ $k =$ ______

50. $AB = 6$, $AC = 10$, $m\angle A = 48°$; $DE = 9$, $DF = 15$, $m\angle D = 48°$. Criterion: ______

51. Same lengths, but the congruent angles are $\angle B$ and $\angle E$. Conclusion: ____________ Why: ____________

52. Sides $12$, $18$, $24$ and $8$, $12$, $16$. Similar? ______ $k =$ ______

53. $\triangle ABC \sim \triangle DEF$, $AB = 14$, $DE = 21$, $BC = 18$. $EF =$ ______

54. $AB = 10$, $AC = 4$, $m\angle A = 72°$; $DE = 25$, $DF = 10$, $m\angle D = 72°$. Criterion ______ $k =$ ______

55. Sides $3$, $4$, $5$ and $6$, $8$, $11$. Similar? ______ Why: ____________

56. Sides $9$, $12$, $15$ and $12$, $16$, $20$. Similar? ______ $k =$ ______

57. $AB = 8$, $AC = 12$, $m\angle A = 62°$; $DE = 12$, $DF = 18$, $m\angle D = 62°$. Criterion ______ $k =$ ______

58. Which arrangement is a similarity criterion but not a congruence criterion? ______

59. Name the three similarity criteria. ____________________

PAGE 14 — Apply and reason · Lesson 7.3

Think It Through

56. Application. Brackets with sides $9$, $12$, $15$ in and $15$, $20$, $25$ in.

    Ratios ______ , ______ , ______ Similar? ______ $k =$ ______

57. Error analysis. A student checks two ratios and writes "similar by SSS." What is missing, and why does it matter?

    ---

58. Reasoning. Why must the angle in SAS similarity be included? Use the SSA failure from Chapter 5.

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PAGE 15 — A proportion

7.4 A Similarity Statement Is a Proportion

FIGURE: fig8-similarity-from-expressions.png (full width)

Fill in the blanks.

Two pairs of corresponding sides give ______ equation.

63. Write the proportion. ____________________

PAGE 16 — Solve, then check twice

Solve, Substitute, Check the Ratio

FIGURE: fig9-solve-the-proportion-then-check.png (full width)

Fill in the blanks.

Check 1: substitute back into ______ expressions.

Check 2: confirm the two ____________ are equal — not the two ____________ .

64. Cross products: ____________________

65. $x =$ ______

66. $AB =$ ______ $DE =$ ______

67. Ratio check: ______ = ______ = ______

68. Why is $x = 2$ not the answer to "find $AB$"? ____________________

PAGE 17 — Practice · proportions

Practice

69. $\tfrac{3}{x} = \tfrac{12}{20}$. $x =$ ______

70. $\tfrac{x}{9} = \tfrac{6}{27}$. $x =$ ______

71. $AB = x$, $DE = 15$, $BC = 8$, $EF = 20$. $x =$ ______

72. $AB = 3x$, $DE = 24$, $AC = 5$, $DF = 20$. $x =$ ______ $AB =$ ______

73. $AB = x + 3$, $DE = 2x$, $BC = 6$, $EF = 9$. $x =$ ______ $AB =$ ______ $DE =$ ______

74. $AB = 2x - 1$, $DE = 3x + 4$, $BC = 5$, $EF = 10$. $x =$ ______ $AB =$ ______ $DE =$ ______

75. $AB = x$, $DE = x + 6$, $BC = 10$, $EF = 4$. $x =$ ______ What it means: ____________________

76. $\tfrac{5}{8} = \tfrac{x}{24}$. $x =$ ______

77. $AB = 4x$, $DE = 18$, $BC = 6$, $EF = 27$. $x =$ ______ $AB =$ ______

78. A solved proportion gives a side length of $-3$. What do you report? ____________________

PAGE 18 — Apply and reason · Lesson 7.4

Think It Through

76. Application. A logo edge is $(3x - 2)$ in; a $\times 3$ enlargement's matching edge is $21$ in.

    $x =$ ______ Print's edge = ______ in

77. Error analysis. $AB = 6$, $DE = 9$, $BC = y$, $EF = 12$; a student writes $\tfrac{6}{9} = \tfrac{12}{y}$ and gets $y = 18$. Error and correction:

    _______________________________________________ Correct $y =$ ______

78. Reasoning. Why is a proportion with a variable still an ordinary equation? What do the cross products do?

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79. Name the two checks that finish an algebraic similarity problem. ____________________

PAGE 19 — SSS from coordinates

7.5 Distance Reaches SSS

FIGURE: fig10-sss-similarity-on-the-coordinate-plane.png (full width)

Fill in the blanks.

Keeping the radicals makes each ratio ____________ on sight.

83. $DE =$ ______ $AB =$ ______

84. The three ratios: ______ , ______ , ______

85. Criterion ______ $k =$ ______

PAGE 20 — AA from slopes

Slope Reaches AA

FIGURE: fig11-slope-gives-aa.png (full width)

Fill in the blanks.

Equal slopes mean the sides are ____________ , which makes the angles between them ____________ .

In Chapter 6, slope certified only ______ . Here it certifies an angle ____________ .

86. Slope of $\overline{AB}$ ______ Slope of $\overline{DE}$ ______

87. What equal slopes prove about the angles: ____________________

88. Criterion reached: ______

PAGE 21 — Practice · coordinates

Practice

89. $A(-7, 1)$, $B(-3, 1)$, $C(-7, 4)$; $D(1, -4)$, $E(9, -4)$, $F(1, 2)$.

    $AB$ ______ $BC$ ______ $AC$ ______ $DE$ ______ $EF$ ______ $DF$ ______ Criterion ______ $k =$ ______

90. $A(-7, 0)$, $B(-3, 2)$, $C(-6, -4)$; $S(2, 2)$, $T(10, 6)$, $U(4, -6)$.

    $AB$ ______ $BC$ ______ $AC$ ______ $ST$ ______ $TU$ ______ $SU$ ______ $k =$ ______

91. $A(0, 0)$, $B(6, 0)$, $C(0, 4)$; $D(-8, 1)$, $E(-2, 1)$, $F(-8, 7)$. Similar? ______ Two ratios: ______ and ______

92. For item 90's triangles: slopes ______ / ______ , ______ / ______ , ______ / ______ What they prove: ____________

93. For item 90's triangles: $k$ from one pair ______ verified with a second pair ______

94. $A(-4, 2)$, $B(0, 4)$, $C(-2, -2)$ dilated by $2$ about the origin.

    Image: ______ , ______ , ______ Three ratios: ______ , ______ , ______

95. Corresponding sides $3\sqrt{7}$ and $5\sqrt{7}$. $k =$ ______

96. Two corresponding sides both have slope $\tfrac{2}{5}$. That establishes: ____________________

97. Which formula reaches SSS similarity? ______ Which reaches AA? ______

PAGE 22 — Apply and reason · Lesson 7.5

Think It Through

95. Error analysis. A student computes $\tfrac{DE}{AB} = \tfrac{3}{2}$ and writes "similar by SSS." What is missing?

    ---

96. Application. Trail map, grid units $100$ m. Loop 1: $(-7, 1)$, $(-3, 1)$, $(-7, 4)$. Loop 2: $(1, -4)$, $(9, -4)$, $(1, 2)$.

    Scaled copy? ______ Loop 1 perimeter ______ m Loop 2 perimeter ______ m Difference ______ m

97. Reasoning. Why does slope reach AA while distance reaches SSS — and why is slope more useful here than in Chapter 6?

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98. Error analysis. A student says three pairs of equal slopes prove the triangles congruent. Correct them.

    ---

99. State the four steps of a coordinate similarity proof. ____________________

PAGE 23 — Dilation, then a rigid motion

7.6 A Dilation and Then Some Moves

FIGURE: fig12-dilation-then-rigid-motion.png (full width)

Fill in the blanks.

A ____________ followed by any sequence of ____________________ carries a figure onto a similar figure.

The dilation changes the ____________ ; the rigid motions change only ____________ .

103. The two steps, in order: ____________________

104. Which step changes the size? ______

105. Which step is a rigid motion? ______

106. $A'$ ______ $B'$ ______ $C'$ ______

PAGE 24 — Order matters

Name the Steps in Order

FIGURE: fig13-naming-the-sequence.png (full width)

107. Do the two orders give the same triangle? ______

108. List everything a complete description of a sequence must contain.

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PAGE 25 — Practice · transformation sequences

Practice

109. $A(-6, 2)$, $B(-2, 4)$, $C(-4, -4)$ dilated by $\tfrac{3}{2}$ about the origin. ______ , ______ , ______

110. Translate item 109's answer $8$ right. ______ , ______ , ______

111. $(2, -3)$ dilated by $4$ about the origin. ______

112. $(-6, 9)$ dilated by $\tfrac{1}{3}$ about the origin. ______

113. $A(1, 1)$, $B(3, 1)$, $C(1, 2)$ onto $D(2, 2)$, $E(6, 2)$, $F(2, 4)$. Sequence: ____________________

114. $A(2, 0)$, $B(4, 0)$, $C(2, 3)$ onto $D(-6, 0)$, $E(-12, 0)$, $F(-6, -9)$. Sequence: ____________________

115. A logo is dilated by $\tfrac{1}{2}$ about the origin, then reflected over the $y$-axis. Image of $(8, -6)$: ______

116. $(-4, 6)$ dilated by $\tfrac{1}{2}$ about the origin. ______

117. The two pieces of information a dilation needs: ______ and ______

118. Do "dilate then translate" and "translate then dilate" always agree? ______

PAGE 26 — Apply and reason · Lesson 7.6

Think It Through

115. Why can rigid motions alone not verify similarity in general?

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116. Error analysis. A student describes a sequence as "a dilation and a translation." List what is missing.

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117. Reasoning. Why does a dilation followed by rigid motions always give a similar figure?

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118. State what an answer to G.TR.3d must contain. ____________________

PAGE 27 — Indirect measurement

7.7 Measuring What You Cannot Reach

FIGURE: fig14-indirect-measurement.png (full width)

Fill in the blanks.

Height goes with ____________ , shadow with ____________ .

Perimeter scales by ______ . Area is Chapter ______ .

123. What makes the two triangles similar? Criterion ______ Both angle pairs: ____________________

124. The proportion: ______ = ______

125. $h =$ ______ ft

126. Why was the tree's height not measured directly? ____________________

127. Are the two triangles drawn to the same scale? ______ What is the same? ______

128. Which angle pair comes from the parallel rays? ____________________

PAGE 28 — Practice · attributes

Practice

129. Person $6$ ft, shadow $8$ ft; pole's shadow $44$ ft. Pole = ______ ft

130. Stick $4$ ft, shadow $3$ ft; building's shadow $51$ ft. Building = ______ ft

131. $k = \tfrac{5}{2}$, perimeter of $\triangle ABC = 22$. Perimeter of $\triangle DEF =$ ______

132. $AB = 8$, $DE = 20$, perimeter of $\triangle ABC = 30$. $k$ ______ Perimeter of $\triangle DEF =$ ______

133. $AB = 9$, $DE = 15$, perimeter of $\triangle DEF = 60$. $k$ ______ Perimeter of $\triangle ABC =$ ______

134. $k = 3$. If $BC = 7$, $EF =$ ______ . If $DF = 27$, $AC =$ ______

135. $\overline{DE} \parallel \overline{BC}$; $AD = 9$, $AB = 24$, $BC = 40$. $DE =$ ______

136. Person $5$ ft, shadow $6$ ft; tree's shadow $42$ ft. Tree = ______ ft

137. $k = 4$, perimeter of $\triangle ABC = 13$. Perimeter of $\triangle DEF =$ ______

PAGE 29 — Apply and reason · Lesson 7.7

Apply It

136. Application. Mirror $30$ ft from a tower; surveyor $4$ ft from the mirror, eyes $5$ ft up.

     Proportion ______ = ______ Tower = ______ ft

137. Application. Scale $1 : 48$; model roof edge $5.5$ in. Real edge ______ in = ______ ft

138. Error analysis. A student writes $\tfrac{5}{36} = \tfrac{h}{4}$ for the shadow figure. Error, correct proportion, and answer:

     ---

139. Reasoning. Why does perimeter scale by $k$, and why does this chapter not make the matching claim about area?

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140. Reasoning. Why do similar triangles let you measure something unreachable? Name the step where the similarity is used.

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141. The criterion behind shadow problems, and the fact about sunlight it needs: ____________________

142. State the four-step method for a measured attribute of a similar pair. ____________________

PAGE 30 — Work frames

Blank Frames

FIGURE: fig15-blank-similarity-frames.png (full width)

Use the ratio table to record three comparisons before naming a criterion, the empty plane for transformation sequences, and the five-row frame for the proof.

Reminder. Fill all three ratio rows before writing a conclusion. Two agreeing ratios prove nothing — the third can still disagree.

PAGE 31 — Chapter review

Review

Review 1 (G.TR.3 a, b). $\overline{DE} \parallel \overline{BC}$, $D$ on $\overline{AB}$, $E$ on $\overline{AC}$. $AD = 2x$, $DB = x + 4$, $\tfrac{AD}{AB} = \tfrac{2}{5}$.

• Two angle congruences with reasons: ____________________ Criterion: ______
• $x =$ ______ $AD$ ______ $DB$ ______ $AB$ ______ Substitution check: ____________
• If $BC = 35$: $DE =$ ______ Reason: ____________
• Why $\tfrac{AD}{DB} = \tfrac{DE}{BC}$ is wrong, and the right proportion: ____________________

Review 2 (G.TR.3 c, d). $A(-7, 0)$, $B(-3, 2)$, $C(-6, -4)$; $S(2, 2)$, $T(10, 6)$, $U(4, -6)$.

• Six lengths: ______ ______ ______ / ______ ______ ______ Criterion ______ $k =$ ______
• Three slope pairs: ______ ______ ______ Criterion reached: ______
• Sequence carrying $\triangle ABC$ onto $\triangle STU$: ____________________
• Why the three are one answer rather than three: ____________________

Review 3 (G.TR.3 e). A $4$ ft rod casts a $5$ ft shadow. At the same moment a cliff's shadow measures $145$ ft on level ground.

• Why the triangles are similar: criterion ______ both angle pairs ____________________
• Proportion ______ = ______ Cliff = ______ ft
• What $\tfrac{4}{145} = \tfrac{h}{5}$ claims, and why it is wrong: ____________________
• $k =$ ______ What it would take to compute either perimeter: ____________________

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.
• Ratio blanks need a fraction's height. Items 41, 47, 49, 59, 84, and 90 are answered with fractions such as $\tfrac{4}{3}$ and with radicals such as $2\sqrt{5}$. Set those blanks at least 1.6 line-heights tall so a stacked fraction and a radical overbar both fit.
• Six-length items (89, 90, Review 2) need six blanks on one line, or a three-by-two grid of blanks — do not let students answer them in a margin.
• Transformation answers are sentences, not values. Items 113, 114, 117, 122, and Review 2 need at least two full ruled lines each; a single short blank invites the incomplete answer item 117 is about.
• The blank plane on page 30 is fig15-blank-similarity-frames.png. Reprint it for any extra coordinate or transformation problem a teacher assigns.
• Symbols: ∼, ≅, ∠, △, ∥, ⊥, √, ′ (prime), and the overline for segments must render in the body font; check the Canva font supports them before the first export. The tilde in $\triangle ABC \sim \triangle DEF$ must not be typed as a swung dash or an approximation sign.
• Item numbers are continuous from 1 to 144 and must not be renumbered when pages are reordered.