Grade 7 Workbook — Chapter 13: Similar Figures and Scale Drawings

SOL 7.MG.2 · Companion to Textbook Chapter 13

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/. Item numbers match the textbook exactly.

PAGE 1 — Chapter opener

Chapter 13 · Similar Figures and Scale Drawings

Standard 7.MG.2

In this chapter you will:

• Use geometric markings to find corresponding congruent angles
• Identify corresponding sides of similar triangles and quadrilaterals
• Write similarity statements with the symbol $\sim$
• Write and solve proportions from corresponding sides
• Justify whether two figures really are similar
• Find missing side lengths and unknown angle measures
• Read and make scale drawings

Words to know: similar · corresponding parts · congruent · scale factor · enlargement · reduction · similarity statement · proportion · scale drawing · scale

Two conditions, both required. Similar figures have corresponding angles that are congruent AND corresponding sides that are proportional. Checking only one of them is the most common mistake in this chapter.

PAGE 2 — What similarity means

13.1 What Similarity Means

FIGURE: fig1-similar-triangles-marked.png (full width)

Fill in the blanks.

Two figures are similar when corresponding angles are ______________ and corresponding sides are ______________.

The number you multiply every side of one figure by to get the matching side of the other is the ______________ ______________.

A scale factor greater than 1 makes an ______________. A scale factor between 0 and 1 makes a ______________.

Congruent figures are the special case of similar figures with a scale factor of ______.

Guided practice.

1. Triangles with sides $3$, $4$, $5$ and $6$, $8$, $10$.

   $\dfrac{6}{3} =$ ______ $\dfrac{8}{4} =$ ______ $\dfrac{10}{5} =$ ______

   Similar? ______ Scale factor, small to large: ______

2. Scale factor, large to small: ______

3. Is a $4$ by $6$ rectangle similar to an $8$ by $10$ rectangle?

   $\dfrac{8}{4} =$ ______ $\dfrac{10}{6} =$ ______ Similar? ______

4. Two triangles each have sides $5$, $12$, $13$, with corresponding angles congruent.

   Similar? ______ Congruent? ______ Scale factor: ______

5. Similar figures have corresponding angles that are ______________ and corresponding sides that are ______________.

PAGE 3 — Both conditions matter

When Figures Are NOT Similar

FIGURE: fig4-not-similar-rectangles.png (full width)

Independent practice.

6. Similar? If so, give the scale factor from the first figure to the second. Show your ratios.

   Pair	Ratios	Similar?	Scale factor
   a) Triangles $6, 9, 12$ and $8, 12, 16$
   b) Triangles $5, 7, 9$ and $10, 14, 20$
   c) Squares with sides $4$ and $7$

7. Triangle with sides $7$, $9$, $11$; scale factor $3$. New sides: ______, ______, ______

8. Triangle with sides $24$, $30$, $36$; scale factor $\tfrac16$. New sides: ______, ______, ______

9. $PQRS$ has sides $10$, $5$, $4$, $5$. $TUVW$ is similar with scale factor $2$.

   Sides of $TUVW$: ______, ______, ______, ______ Scale factor $TUVW$ back to $PQRS$: ______

10. Is a $3$ by $5$ rectangle similar to a $9$ by $15$ rectangle? Ratios: ______ and ______ Scale factor: ______

11. Application. A $4$ in by $6$ in photo is enlarged to a width of $10$ in.

    Scale factor: ______ New length: ______

12. Reasoning. Jamal says any two rectangles are similar because all their angles are right angles. Give a counterexample and name the condition he forgot.

    Counterexample: ____________________________________________

    Missing condition: __________________________________________

PAGE 4 — Exit ticket 13.1

Exit Ticket · Lesson 13.1

Name: ________________________ Date: ____________

13. Triangles $4$, $6$, $8$ and $6$, $9$, $12$ with congruent corresponding angles. Similar? ______ Scale factor: ______

14. Triangle $9$, $12$, $15$ with scale factor $\tfrac23$. New sides: ______, ______, ______

15. Is a $2$ by $5$ rectangle similar to a $4$ by $8$ rectangle? Ratios: ______ and ______ Similar? ______

16. Why is "same shape, different size" not a complete definition of similar?

PAGE 5 — Matching the parts

13.2 Corresponding Angles and Sides

FIGURE: fig1-similar-triangles-marked.png (full width)

Complete the correspondence table for $\triangle ABC \sim \triangle DEF$.

Matching arcs mean the angles are congruent. Matching hash marks mean the sides correspond — they are proportional, not equal.

Guided practice.

17. Angle of $\triangle DEF$ congruent to $\angle B$: ______

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

19. For $PQRS \sim TUVW$: angle corresponding to $\angle R$ is ______; side corresponding to $\overline{RS}$ is ______

20. $\triangle ABC \sim \triangle DEF$, $m\angle A = 50°$, $m\angle B = 60°$.

    $m\angle C =$ ______ $m\angle D =$ ______ $m\angle E =$ ______ $m\angle F =$ ______

21. $\triangle GHJ \sim \triangle KLM$: $\angle H \cong$ ______ and $\overline{GJ}$ corresponds to ______

PAGE 6 — Angle measures carry across

Angles Do Not Change When You Scale

FIGURE: fig6-angle-measures-carry.png (full width)

Remember: triangle angles add to $180°$. Quadrilateral angles add to $360°$.

Independent practice.

22. $\triangle RST \sim \triangle XYZ$.

    Angles: $\angle R \cong$ ______, $\angle S \cong$ ______, $\angle T \cong$ ______

    Sides: $\overline{RS} \leftrightarrow$ ______, $\overline{ST} \leftrightarrow$ ______, $\overline{TR} \leftrightarrow$ ______

23. $ABCD \sim EFGH$: angle corresponding to $\angle C$ is ______; side corresponding to $\overline{DA}$ is ______

24. $\triangle ABC \sim \triangle DEF$, $m\angle D = 95°$, $m\angle E = 35°$.

    $m\angle F =$ ______ $m\angle A =$ ______ $m\angle B =$ ______ $m\angle C =$ ______

25. $PQRS \sim TUVW$, $m\angle P = 110°$, $m\angle Q = 70°$, $m\angle R = 110°$.

    $m\angle S =$ ______ $m\angle T =$ ______ $m\angle U =$ ______ $m\angle V =$ ______ $m\angle W =$ ______

26. True or false: in $\triangle ABC \sim \triangle DEF$, $\overline{AB}$ corresponds to $\overline{EF}$. ______

    If false, the correct corresponding side is ______

27. Application. Two similar sails, with $\angle P \cong \angle W$, $\angle Q \cong \angle X$, $\angle R \cong \angle Y$.

    Side corresponding to $\overline{QR}$: ______ If $m\angle P = 42°$ and $m\angle Q = 108°$, then $m\angle Y =$ ______

28. Reasoning. Why do two angle measures in one triangle determine all six angle measures in a similar pair?

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PAGE 7 — Exit ticket 13.2

Exit Ticket · Lesson 13.2

Name: ________________________ Date: ____________

29. In $\triangle ABC \sim \triangle DEF$, the angle corresponding to $\angle C$ is ______

30. In $\triangle ABC \sim \triangle DEF$, the side corresponding to $\overline{BC}$ is ______

31. $\triangle ABC \sim \triangle DEF$, $m\angle A = 64°$, $m\angle F = 81°$. Then $m\angle B =$ ______

32. How do matching arcs on two angles tell you which parts correspond?

PAGE 8 — The order carries the meaning

13.3 Writing Similarity Statements

FIGURE: fig2-correspondence-map.png (full width)

Stack the two names and read down each column.

So $\angle A \cong$ ______, $\angle B \cong$ ______, $\angle C \cong$ ______

and $\overline{AB} \leftrightarrow$ ______, $\overline{BC} \leftrightarrow$ ______, $\overline{CA} \leftrightarrow$ ______

Guided practice.

33. $\angle J \cong \angle X$, $\angle K \cong \angle Y$, $\angle L \cong \angle Z$ → ______________________

34. $\angle A \cong \angle F$, $\angle B \cong \angle D$, $\angle C \cong \angle E$ → ______________________

35. Rewrite $\triangle ABC \sim \triangle DEF$ beginning with $C$: ______________________

36. $\angle P \cong \angle T$, $\angle Q \cong \angle U$, $\angle R \cong \angle V$, $\angle S \cong \angle W$ → ______________________

37. $\triangle MNP \sim \triangle QRS$: $\angle N \cong$ ______ and $\overline{MP}$ corresponds to ______

PAGE 9 — Statement practice

Writing and Fixing Statements

Independent practice.

38. Write each similarity statement.

    a) $\angle D \cong \angle P$, $\angle E \cong \angle Q$, $\angle F \cong \angle R$ → ______________________

    b) $\angle G \cong \angle M$, $\angle H \cong \angle K$, $\angle J \cong \angle L$ → ______________________

39. $\angle A \cong \angle W$, $\angle B \cong \angle X$, $\angle C \cong \angle Y$, $\angle D \cong \angle Z$ → ______________________

40. Two other correct forms of $\triangle ABC \sim \triangle DEF$:

    ______________________ and ______________________

41. A student writes $\triangle RST \sim \triangle UVW$, but the markings show $\angle R \cong \angle V$, $\angle S \cong \angle U$, $\angle T \cong \angle W$.

    Correct statement: ______________________

42. $\triangle ABC \sim \triangle DEF$, $AB = 6$, $DE = 9$. Ratio from $\triangle ABC$ to $\triangle DEF$ in lowest terms: ______

43. Application. Pennants with $\angle L \cong \angle T$, $\angle M \cong \angle U$, $\angle N \cong \angle S$ → ______________________

44. Reasoning. What information is lost if you only say "these two triangles are similar"?

    ---

    A question you could not answer: _______________________________________________

PAGE 10 — Exit ticket 13.3

Exit Ticket · Lesson 13.3

Name: ________________________ Date: ____________

45. $\angle A \cong \angle R$, $\angle B \cong \angle S$, $\angle C \cong \angle T$ → ______________________

46. $\triangle PQR \sim \triangle XYZ$: side corresponding to $\overline{PR}$ is ______

47. Rewrite $\triangle PQR \sim \triangle XYZ$ beginning with $Q$: ______________________

48. Why do $\triangle ABC \sim \triangle DEF$ and $\triangle ABC \sim \triangle EDF$ say different things?

PAGE 11 — Proportion frames

13.4 Proportions from Corresponding Sides

FIGURE: fig3-similar-quadrilaterals.png (full width)

Fill the proportion frames.

For $\triangle ABC \sim \triangle DEF$:

$$\frac{AB}{\rule{1.2cm}{0.4pt}} = \frac{\rule{1.2cm}{0.4pt}}{EF} = \frac{CA}{\rule{1.2cm}{0.4pt}}$$

For $PQRS \sim TUVW$:

$$\frac{PQ}{\rule{1.2cm}{0.4pt}} = \frac{QR}{\rule{1.2cm}{0.4pt}} = \frac{RS}{\rule{1.2cm}{0.4pt}} = \frac{SP}{\rule{1.2cm}{0.4pt}}$$

Check the trapezoids above: $\dfrac{15}{10} =$ ______ $\dfrac{7.5}{5} =$ ______ $\dfrac{6}{4} =$ ______ Scale factor: ______

Guided practice.

49. Three equal ratios for $\triangle ABC \sim \triangle DEF$: _______________________________________________

50. Triangles $6, 9, 12$ and $10, 15, 20$: ratios ______, ______, ______ Similar? ______

51. Quadrilaterals $4, 6, 8, 10$ and $8, 12, 16, 18$: ratios ______, ______, ______, ______ Similar? ______

52. Rectangles $4$ by $6$ and $8$ by $10$: ratios ______ and ______ Similar? ______

    Why were congruent angles not enough? _______________________________________________

53. A proportion using only $AB$, $DE$, $BC$, $EF$: _______________________

PAGE 12 — Similar or not?

Justify With Ratios

Independent practice.

54. Complete the table.

    Pair	Ratios	Similar?
    a) Triangles $5, 6, 7$ and $15, 18, 21$
    b) Triangles $8, 10, 12$ and $12, 15, 20$
    c) Quadrilaterals $3, 4, 5, 6$ and $9, 12, 15, 18$

55. Three equal ratios for $\triangle PQR \sim \triangle STU$: _______________________________________________

56. $\triangle ABC \sim \triangle DEF$, $AB = 5$, $BC = 8$, $DE = 15$.

    Proportion for $EF$: _______________________ Scale factor $\triangle ABC \to \triangle DEF$: ______

57. A student writes $\dfrac{AB}{EF} = \dfrac{BC}{DE}$ for $\triangle ABC \sim \triangle DEF$.

    Error: _______________________________________________ Correct proportion: _______________________

58. $ABCD$ has sides $6$, $9$, $12$, $15$; $EFGH$ is similar with scale factor $\tfrac23$.

    Sides of $EFGH$: ______, ______, ______, ______

59. Application. Pool $24$ ft by $36$ ft; model $8$ in by $12$ in.

    $\dfrac{8}{24} =$ ______ $\dfrac{12}{36} =$ ______ Same shape? ______

60. Reasoning. Square with side $4$ and rhombus with side $8$ whose angles are $60°$ and $120°$.

    All four side ratios: ______ Similar? ______ Condition that fails: ______________________

PAGE 13 — Exit ticket 13.4

Exit Ticket · Lesson 13.4

Name: ________________________ Date: ____________

61. Three equal ratios for $\triangle ABC \sim \triangle DEF$: _______________________________________________

62. Triangles $4, 5, 6$ and $12, 15, 18$: ratios ______, ______, ______ Similar? ______

63. Triangles $6, 8, 10$ and $9, 12, 16$: ratios ______, ______, ______ Similar? ______

64. Why do proportional sides alone not guarantee similar quadrilaterals?

PAGE 14 — Solving for a missing side

13.5 Finding a Missing Side Length

FIGURE: fig5-missing-side-setup.png (full width)

The frame. Write the pair you know completely on the left, the pair with the unknown on the right.

$$\frac{\text{known side of figure 1}}{\text{its partner}} = \frac{\text{side of figure 1}}{\text{unknown partner}}$$

Guided practice.

65. $\triangle ABC \sim \triangle DEF$, $AB = 6$, $DE = 9$, $BC = 8$.

    $\dfrac{6}{9} = \dfrac{8}{\rule{0.9cm}{0.4pt}}$ Cross products: ______ $=$ ______ $EF =$ ______

66. Same triangles, $CA = 10$. $FD =$ ______

67. $\triangle ABC \sim \triangle DEF$, $AB = 10$, $DE = 4$, $CA = 15$. $FD =$ ______

68. $PQRS \sim TUVW$, $PQ = 12$, $TU = 8$, $RS = 9$. $VW =$ ______

69. $\triangle ABC \sim \triangle DEF$, $AB = 5$, $DE = 12.5$. Scale factor $\triangle ABC \to \triangle DEF$: ______

PAGE 15 — Missing-side practice

Solve and Check

Independent practice.

70. Solve each.

    Problem	Proportion	Answer
    a) $\triangle ABC \sim \triangle DEF$, $AB=4$, $DE=10$, $BC=6$; find $EF$
    b) $\triangle ABC \sim \triangle DEF$, $CA=9$, $FD=6$, $AB=12$; find $DE$
    c) $ABCD \sim EFGH$, $AB=15$, $EF=5$, $CD=21$; find $GH$

71. Scale factor $\tfrac34$ from $\triangle ABC$ to $\triangle DEF$, with $AB=16$, $BC=20$, $CA=28$.

    $DE =$ ______ $EF =$ ______ $FD =$ ______

72. From the figure on page 14: $x =$ ______ $y =$ ______

73. $\triangle ABC \sim \triangle DEF$, $AB=7$, $DE=10$, $BC=9$. $EF \approx$ ______ (nearest tenth)

74. $JKLM \sim NPQR$, $JK=18$, $NP=12$, $QR=10$. $LM =$ ______

75. Application. A $6$ ft person casts a $4$ ft shadow; a flagpole casts a $22$ ft shadow.

    Proportion: _______________________ Flagpole height: ______

76. Reasoning. Why is this the same skill as solving a proportion in Chapter 5, and how do you choose which two ratios to set equal?

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PAGE 16 — Exit ticket 13.5

Exit Ticket · Lesson 13.5

Name: ________________________ Date: ____________

77. $\triangle ABC \sim \triangle DEF$, $AB=5$, $DE=15$, $BC=7$. $EF =$ ______

78. $ABCD \sim EFGH$, $BC=24$, $FG=9$, $AB=32$. $EF =$ ______

79. $\triangle ABC \sim \triangle DEF$, $CA=11$, $FD=4$. Scale factor $\triangle DEF \to \triangle ABC$: ______

80. $\triangle ABC \sim \triangle DEF$, $AB=9$, $DE=6$, $BC=5$. $EF \approx$ ______ (nearest tenth)

PAGE 17 — Reading a scale drawing

13.6 Scale Drawings

FIGURE: fig7-floor-plan.png (full width)

The scale frame.

$$\frac{1 \text{ drawing unit}}{\text{actual units it represents}} = \frac{\text{drawing length}}{\text{actual length}}$$

Guided practice.

81. Scale $1 \text{ in} = 6 \text{ ft}$; wall measures $3.5$ in.

    $\dfrac{1}{6} = \dfrac{3.5}{\rule{0.9cm}{0.4pt}}$ Actual length: ______ ft

82. Same scale; hallway is actually $45$ ft. Length on the plan: ______ in

83. Map scale $1 \text{ in} = 20 \text{ mi}$; towns $4.5$ in apart. Actual distance: ______ mi

84. Grid where each square $= 3$ ft; room is $5$ squares by $8$ squares. Actual: ______ ft by ______ ft

85. Scale $1 : 12$; real chair $48$ in tall. Model height: ______ in

PAGE 18 — Grid work and scale practice

Working On a Grid

FIGURE: fig8-scale-drawing-grid.png (full width)

Independent practice.

86. Plan scale $1 \text{ in} = 8 \text{ ft}$.

    a) Bedroom $1.5$ in by $2$ in → ______ ft by ______ ft

    b) Closet $0.5$ in wide → ______ ft

87. Map scale $1 \text{ cm} = 15 \text{ km}$; cities $7.4$ cm apart → ______ km

88. Drawing scale $1 \text{ in} = 4 \text{ ft}$; room is $18$ ft long → ______ in on the drawing

89. Model scale $1 : 12$; real car $180$ in long → ______ in

90. Using the floor plan on page 17:

    Living room: ______ ft by ______ ft Kitchen: ______ ft by ______ ft

91. Application. Garden $30$ ft by $42$ ft at $1 \text{ in} = 12 \text{ ft}$.

    Draw a rectangle ______ in by ______ in.

    Draw it on the grid below. Each grid square = $\tfrac12$ in.

    BLANK GRID: 8 columns by 8 rows of half-inch squares

92. Reasoning. A student says $1 \text{ in} = 50 \text{ mi}$ means the map is bigger than the land. Explain the error.

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PAGE 19 — Exit ticket 13.6

Exit Ticket · Lesson 13.6

Name: ________________________ Date: ____________

93. Scale $1 \text{ in} = 5 \text{ ft}$; wall is $6.5$ in on the plan → ______ ft

94. Scale $1 \text{ cm} = 25 \text{ km}$; points $3.2$ cm apart → ______ km

95. Scale $1 : 12$; real desk $60$ in wide → ______ in

96. Why must the scale be stated on every scale drawing?

PAGE 20 — Chapter 13 review, part 1

Chapter 13 Review

Part A · Corresponding congruent angles (7.MG.2a)

97. From fig1-similar-triangles-marked.png, the three pairs of corresponding congruent angles:

    ______ $\cong$ ______ ______ $\cong$ ______ ______ $\cong$ ______

98. From fig3-similar-quadrilaterals.png, the four pairs:

    ______ $\cong$ ______ ______ $\cong$ ______ ______ $\cong$ ______ ______ $\cong$ ______

99. $\angle B$ and $\angle E$ each carry a double arc. What does that tell you? In symbols: ______________

Part B · Corresponding sides (7.MG.2b)

100. $\triangle ABC \sim \triangle DEF$: side corresponding to $\overline{CA}$ is ______

101. $ABCD \sim EFGH$: side corresponding to $\overline{BC}$ is ______

102. In the trapezoid figure, side of $TUVW$ corresponding to $\overline{PQ}$ is ______; lengths ______ and ______

Part C · Similarity statements (7.MG.2c)

103. $\angle M \cong \angle B$, $\angle N \cong \angle C$, $\angle P \cong \angle A$ → ______________________

104. $\angle J \cong \angle R$, $\angle K \cong \angle S$, $\angle L \cong \angle T$, $\angle M \cong \angle U$ → ______________________

105. $\triangle ABC \sim \triangle DEF$ beginning with $B$: ______________________

Part D · Proportions from corresponding sides (7.MG.2d)

106. Three equal ratios for $\triangle ABC \sim \triangle DEF$: _______________________________________________

107. $PQRS \sim TUVW$: a proportion using $PQ$, $TU$, $QR$, $UV$: _______________________

108. $\triangle ABC \sim \triangle DEF$, $AB=6$, $DE=15$, $BC=8$.

     Proportion: _______________________ Scale factor: ______

PAGE 21 — Chapter 13 review, part 2

Chapter 13 Review (continued)

Part E · Justifying similarity with ratios (7.MG.2e)

109. Triangles $9, 12, 15$ and $12, 16, 20$: ratios ______, ______, ______ Similar? ______

110. Quadrilaterals $5, 7, 9, 11$ and $10, 14, 18, 20$: ratios ______, ______, ______, ______ Similar? ______

111. Rectangles $6$ by $9$ and $10$ by $15$: ratios ______ and ______ Similar? ______

Part F · Missing side lengths (7.MG.2f)

112. $\triangle ABC \sim \triangle DEF$, $AB=12$, $DE=8$, $CA=21$. $FD =$ ______

113. $ABCD \sim EFGH$, $AB=7$, $EF=17.5$, $BC=6$. $FG =$ ______

114. $\triangle ABC \sim \triangle DEF$, $AB=5$, $DE=9$, $BC=7$. $EF \approx$ ______ (nearest tenth)

Part G · Unknown angle measures (7.MG.2g)

115. $\triangle ABC \sim \triangle DEF$, $m\angle A = 38°$, $m\angle B = 97°$.

     $m\angle C =$ ______ $m\angle D =$ ______ $m\angle E =$ ______ $m\angle F =$ ______

116. $ABCD \sim EFGH$, $m\angle A = 85°$, $m\angle B = 95°$, $m\angle C = 120°$.

     $m\angle D =$ ______ $m\angle E =$ ______ $m\angle F =$ ______ $m\angle G =$ ______ $m\angle H =$ ______

Part H · Scale drawings (7.MG.2h)

117. Scale $1 \text{ in} = 9 \text{ ft}$; room $3.5$ in by $2$ in → ______ ft by ______ ft

118. Scale $1 \text{ cm} = 40 \text{ km}$; cities $6.5$ cm apart → ______ km

PAGE 22 — Chapter 13 review, part 3

Chapter 13 Review (continued)

Part I · Mixed application and reasoning

119. Application. A $5$ ft student casts a $3$ ft shadow; a tree casts a $27$ ft shadow.

     Proportion: _______________________ Tree height: ______ ft

120. Application. Blueprint scale $1 \text{ in} = 4 \text{ ft}$; deck measures $3$ in by $2.5$ in.

     Actual dimensions: ______ ft by ______ ft Actual area: ______ square feet

121. Reasoning. Why are a $4$ by $6$ rectangle and an $8$ by $10$ rectangle not similar, even though all their angles are congruent?

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122. Reasoning. Why does the order of the letters matter? Use $\triangle ABC \sim \triangle DEF$ and $\triangle ABC \sim \triangle EDF$.

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Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Figure widths: fig1-similar-triangles-marked.png, fig3-similar-quadrilaterals.png, fig5-missing-side-setup.png, and fig6-angle-measures-carry.png are wide landscape art — place full width. fig4-not-similar-rectangles.png, fig7-floor-plan.png, and fig8-scale-drawing-grid.png are closer to square; place at about 70 percent width, centered.
• Blank grid on page 18: an 8 × 8 grid of half-inch squares, drawn as a Canva table with equal row and column sizes so the squares stay square when the page is resized
• Proportion frames: keep each fraction bar and its blank inside one text box so the layout does not reflow
• Correspondence tables: set a fixed row height so students have room to write full segment names such as $\overline{DE}$
• Answer blanks: keep every blank on the same line as its prompt