Appendix A — Answer Key, Chapter 14: Quadrilaterals and Their Properties

SOL 7.MG.3 · Covers textbook Chapter 14 and the companion workbook. Item numbers match the textbook; workbook items are the same problems with the same numbers, so this key serves both. Reasoning answers show an acceptable response, not the only wording.

Trapezoid convention. This chapter uses the exclusive definition — a trapezoid has exactly one pair of parallel sides — so no parallelogram, rectangle, rhombus, or square is a trapezoid. Items 20, 23, 31, 76, and 79 are the only items whose answers would change under the inclusive definition.

Lesson 14.1 — The Five Named Quadrilaterals and Their Properties

Guided practice

1. $\overline{AB} \parallel \overline{DC}$ and $\overline{BC} \parallel \overline{AD}$; $\overline{AB} \cong \overline{DC}$ and $\overline{BC} \cong \overline{AD}$. The arrowheads give the parallel pairs and the hash marks give the congruent pairs.
2. Four right angles. $\overline{EF} \cong \overline{HG}$ and $\overline{FG} \cong \overline{EH}$ (opposite sides of a rectangle are congruent).
3. Any three of: all four sides congruent (single hash mark on each); four right angles (a square in each corner); both pairs of opposite sides parallel; opposite sides congruent.
4. All four sides are congruent — that is the defining property of a rhombus. The four angles are not all congruent unless the rhombus is a square; a rhombus only guarantees that opposite angles are congruent. Congruent sides and congruent angles are separate requirements.
5. $\overline{ST} \parallel \overline{VU}$; the congruent legs are $\overline{TU}$ and $\overline{VS}$.

Independent practice

"No" means "not always." A rectangle's diagonals are perpendicular only when it is a square, and a rhombus's diagonals are congruent only when it is a square.

7. Square $4$; rectangle that is not a square $2$; rhombus that is not a square $2$; parallelogram that is neither $0$; isosceles trapezoid $1$.
8. Rectangle and square.
9. Rhombus and square.
10. $90°$. The diagonals of a rhombus are perpendicular, so they meet at a right angle.
11. The frame is not a rectangle. In a rectangle the diagonals are congruent, so two different diagonal measurements prove at least one corner is not a right angle — the frame is out of square and needs adjusting.
12. Folding across a midline sends one slanted side onto the other but they lean opposite ways, so the halves do not match. Folding across a diagonal fails too, because the two halves of a non-rhombus parallelogram have different side lengths. The figure does look balanced, but that balance comes from turning it a half turn, and rotation is not reflection.

Exit ticket 14.1

13. Rhombus and square.
14. $2$ lines. They are the two midlines — the lines through the midpoints of each pair of opposite sides. A rectangle does not fold across its diagonals.
15. False. In a parallelogram the diagonals bisect each other; they are congruent only when the parallelogram is a rectangle (which includes squares).
16. Parallel is about direction: the two sides run the same way and never meet. Congruent is about length: the two sides measure the same. They are independent facts. In a parallelogram, each pair of opposite sides is both parallel and congruent — for example $\overline{AB}$ and $\overline{DC}$.

Lesson 14.2 — Sorting and Classifying Quadrilaterals

Guided practice

17. True. A rectangle requires four right angles, and a square has four right angles.
18. False. A square requires four congruent sides, and a rectangle need not have them — a $3$ by $8$ rectangle is a counterexample.
19. True. A rhombus is defined as a parallelogram with four congruent sides, so it already has both pairs of opposite sides parallel.
20. No. A trapezoid has exactly one pair of parallel sides, and a parallelogram has two pairs, so a parallelogram fails the requirement.
21. Square.

Independent practice

22. a) Rectangle (it is not a square, since $5 \neq 9$) b) Rhombus c) Isosceles trapezoid d) Parallelogram — congruent diagonals would make it a rectangle and perpendicular diagonals would make it a rhombus, and it has neither
23. Quadrilateral, parallelogram, rectangle, and rhombus. Not a trapezoid, since a square has two pairs of parallel sides rather than exactly one.
24. A rectangle that is not a square, whose two lines are the midlines; and a rhombus that is not a square, whose two lines are the diagonals.
25. Rhombus (specifically one that is not a square, since a square's diagonals are congruent).
26. Rectangle (specifically one that is not a square, since a square's diagonals are perpendicular).
27. Rhombus. Four congruent sides plus perpendicular diagonals fit a rhombus. The diagonals being different lengths rules out a square, because a square's diagonals are congruent — so the tile has no right angles at its corners.
28. She is right about any rhombus that is not a square: a rhombus with angles $60°$, $120°$, $60°$, $120°$ has four congruent sides and no right angle, so it is not a rectangle. She is wrong about a square — a square with side $5$ is a rhombus, and it also has four right angles, so it is a rectangle.

Exit ticket 14.2

29. Square.
30. No. A parallelogram with sides $4$, $7$, $4$, $7$ has both pairs of opposite sides parallel but not four congruent sides.
31. No. In this book a trapezoid has exactly one pair of parallel sides while a parallelogram has two, and a figure cannot have both exactly one pair and two pairs.
32. Because most quadrilaterals have several correct names at once — a square is truthfully a square, a rectangle, a rhombus, a parallelogram, and a quadrilateral. Asking "what shape is this?" has many right answers, so asking for the most specific name is the only version of the question with a single answer.

Lesson 14.3 — Finding Unknown Angle Measures

Guided practice

33. $85 + 100 + 70 = 255$, so $x = 360° - 255° = 105°$. Property: the angles of a quadrilateral sum to $360°$.
34. $x = m\angle B = 180° - 62° = 118°$ (consecutive angles of a parallelogram are supplementary); $y = m\angle C = 62°$ (opposite angles are congruent); $z = m\angle D = 118°$ (opposite angles are congruent). Check: $62 + 118 + 62 + 118 = 360$.
35. $n = m\angle S = 180° - 63° = 117°$, because $\angle P$ and $\angle S$ lie on the same leg between the parallel bases and are therefore supplementary. The trapezoid is isosceles, so $m\angle Q = 63°$ and $m\angle R = 117°$. Check: $63 + 63 + 117 + 117 = 360$.
36. $112 + 68 + 95 = 275$, so the fourth angle is $360° - 275° = 85°$.
37. The other three all measure $90°$: the consecutive angle is $180° - 90° = 90°$, and opposite angles are congruent. Most specific name: rectangle (a square only if all four sides are also congruent).

Independent practice

38. a) $360 - (100 + 80 + 100) = 360 - 280 = 80°$ b) $360 - (55 + 125 + 55) = 360 - 235 = 125°$ c) $360 - (90 + 90 + 130) = 360 - 310 = 50°$
39. $m\angle B = 180° - 47° = 133°$ (consecutive angles supplementary); $m\angle C = 47°$ and $m\angle D = 133°$ (opposite angles congruent). Check: $47 + 133 + 47 + 133 = 360$.
40. $\angle A$ and $\angle B$ are consecutive, so they are supplementary: $(3x + 5) + 2x = 180$, giving $5x + 5 = 180$, $5x = 175$, $x = 35$. Then $m\angle A = 3(35) + 5 = 110°$ and $m\angle B = 2(35) = 70°$. Check: $110 + 70 = 180$.
41. The opposite angle also measures $108°$, and the two consecutive angles each measure $180° - 108° = 72°$. So the four are $108°$, $72°$, $108°$, $72°$. A rhombus is a parallelogram — it has both pairs of opposite sides parallel — so every parallelogram angle rule applies to it.
42. The other base angle is also $74°$ (base angles of an isosceles trapezoid are congruent), and each angle at the other base is $180° - 74° = 106°$. Check: $74 + 74 + 106 + 106 = 360$.
43. $360° - 90° - 90° - 68° = 112°$. Property: the four angles of a quadrilateral sum to $360°$.
44. In a parallelogram, opposite angles are congruent and consecutive angles are supplementary. One measure therefore gives its opposite immediately and both neighbors by subtracting from $180°$. A general quadrilateral has no such relationships — its only rule is that the four measures total $360°$ — so one measure leaves the remaining three free to be almost anything that adds to what is left.

Exit ticket 14.3

45. $78 + 102 + 61 = 241$, so the fourth angle is $360° - 241° = 119°$.
46. $m\angle S = 115°$ (opposite angles congruent); $m\angle P = 180° - 115° = 65°$ and $m\angle R = 65°$ (consecutive angles supplementary). Check: $65 + 115 + 65 + 115 = 360$.
47. Each measures $180° - 82° = 98°$, because each is on a leg with an $82°$ base angle and same-leg angles are supplementary.
48. Drawing one diagonal cuts the quadrilateral into two triangles that do not overlap and together make up the whole figure. Each triangle's angles total $180°$, and together the two triangles' angles account for exactly the quadrilateral's four angles, so the total is $180° + 180° = 360°$.

Lesson 14.4 — Finding Unknown Side Lengths

Guided practice

49. Opposite sides of a parallelogram are congruent, so $2x + 3 = 15$, $2x = 12$, $x = 6$. Also $\overline{AD}$ is opposite $\overline{BC}$, so $m = AD = 9$.
50. All four sides of a rhombus are congruent, so $3y - 1 = 11$, $3y = 12$, $y = 4$. Each side measures $11$, so the perimeter is $4 \times 11 = 44$.
51. The diagonals of a rectangle bisect each other and are congruent, so all four half-diagonals are congruent: $p = 7$. Then $JL = 7 + 7 = 14$.
52. Opposite sides of a parallelogram are congruent, so the perimeter is $2(13) + 2(8) = 26 + 16 = 42$.
53. All four sides of a square are congruent, so each side is $52 \div 4 = 13$.

Independent practice

54. $\overline{AB}$ and $\overline{DC}$ are opposite sides of a parallelogram, so $5x = 35$, $x = 7$, and $AB = 35$.
55. All four sides of a rhombus are congruent, so $4t + 3 = 23$, $4t = 20$, $t = 5$. Each side measures $23$, so the perimeter is $4 \times 23 = 92$.
56. The diagonals of a parallelogram bisect each other. From $MD = 18$ we get $BM = 18$, so $6x = 18$ and $x = 3$. Then $BD = 18 + 18 = 36$, and since $AM = 9$ we also get $MC = 9$ and $AC = 9 + 9 = 18$.
57. The diagonals of a rectangle are congruent, so $5n = 30$, $n = 6$, and $AC = 30$. The diagonals also bisect each other, so one half-diagonal measures $30 \div 2 = 15$.
58. The legs of an isosceles trapezoid are congruent, so $3w = 21$ and $w = 7$. Each leg measures $21$, so the perimeter is $26 + 14 + 21 + 21 = 82$.
59. $91$ in. The two diagonals of a rectangle are congruent, so the second brace matches the first exactly.
60. A rhombus has four congruent sides, so one side length $s$ gives every side and the perimeter is $4s$. A parallelogram only guarantees that opposite sides are congruent, so one side length tells you about two of the four sides; the other pair could be any length, and without it the perimeter is undetermined.

Exit ticket 14.4

61. Opposite sides are congruent: $3x + 4 = 22$, $3x = 18$, $x = 6$.
62. $4 \times 9 = 36$.
63. The diagonals of a rectangle bisect each other, so $AC = 11 + 11 = 22$; they are also congruent, so $BD = AC = 22$.
64. $\overline{AD}$ and $\overline{BC}$ are opposite sides of the parallelogram, and opposite sides of a parallelogram are congruent. So $AD = BC = 14$ without any measuring.

Chapter 14 Review

Part A — Comparing and contrasting properties (7.MG.3a)

65. Parallelogram, rectangle, rhombus, and square.
66. Rhombus and square.
67. Rectangle and square.
68. All four: the diagonals of a parallelogram bisect each other, and rectangles, rhombuses, and squares are all parallelograms.
69. Rectangle and square.
70. Rhombus and square.
71. Square $4$; rectangle that is not a square $2$; rhombus that is not a square $2$; parallelogram that is neither $0$; isosceles trapezoid $1$.
72. Any two of: all four sides congruent; perpendicular diagonals; two lines of symmetry (along the diagonals); each diagonal bisects a pair of opposite angles.

Part B — Sorting and classifying (7.MG.3b)

73. Square.
74. Rectangle. Bisecting, congruent diagonals give a rectangle, and the unequal side lengths $8$ and $5$ rule out a square.
75. Isosceles trapezoid.
76. Quadrilateral, parallelogram, rectangle, and rhombus. Not trapezoid, under this book's exclusive definition.
77. False. A rectangle requires four right angles, and a parallelogram with angles $70°$, $110°$, $70°$, $110°$ has none.
78. True. A rhombus requires four congruent sides, and a square has four congruent sides.
79. No. A rectangle has two pairs of parallel sides, and a trapezoid must have exactly one pair.
80. Rhombus (one that is not a square, since a square also has congruent diagonals).

Part C — Unknown angle measures (7.MG.3c)

81. $96 + 84 + 113 = 293$, so the fourth angle is $360° - 293° = 67°$.
82. $m\angle B = 180° - 54° = 126°$ (consecutive angles supplementary); $m\angle C = 54°$ and $m\angle D = 126°$ (opposite angles congruent). Check: $54 + 126 + 54 + 126 = 360$.
83. The other base angle is $71°$ (base angles of an isosceles trapezoid are congruent), and each angle at the other base is $180° - 71° = 109°$. So the other three are $71°$, $109°$, $109°$. Check: $71 + 71 + 109 + 109 = 360$.
84. Consecutive angles of a parallelogram are supplementary: $4x + 5x = 180$, so $9x = 180$ and $x = 20$. The angles measure $4(20) = 80°$ and $5(20) = 100°$. Check: $80 + 100 = 180$.
85. A rhombus is a parallelogram, so the opposite angle is $124°$ and each consecutive angle is $180° - 124° = 56°$. The other three are $56°$, $124°$, $56°$. Check: $124 + 56 + 124 + 56 = 360$.
86. $360° - 90° - 90° - 55° = 125°$.

Part D — Unknown side lengths (7.MG.3d)

87. Opposite sides of a parallelogram are congruent: $4x - 5 = 27$, $4x = 32$, $x = 8$.
88. All four sides of a rhombus are congruent: $6y = 30$, $y = 5$. Each side measures $30$, so the perimeter is $4 \times 30 = 120$.
89. The diagonals of a parallelogram bisect each other, so $AM = MC$: $2z = 14$, $z = 7$. Then $AM = 14$ and $AC = 14 + 14 = 28$.
90. The diagonals of a rectangle are congruent, so $BD = 34$. They also bisect each other, so $AM = 34 \div 2 = 17$.
91. The legs of an isosceles trapezoid are congruent: $5k = 35$, $k = 7$. Each leg measures $35$, so the perimeter is $40 + 24 + 35 + 35 = 134$.

Part E — Mixed application and reasoning

92. All four sides of a rhombus are congruent, so the perimeter is $4 \times 7 = 28$ in. A rhombus is a parallelogram, so the angle opposite the $116°$ angle is also $116°$, and each consecutive angle is $180° - 116° = 64°$. The four angles are $116°$, $64°$, $116°$, $64°$. Check: $116 + 64 + 116 + 64 = 360$.
93. The other diagonal should measure $80$ in, because the diagonals of a rectangle are congruent. A reading of $78$ in would mean the frame is not actually a rectangle — its corners are not all right angles, so it is out of square and needs to be adjusted before installation.
94. A rectangle requires four right angles; a square requires four right angles and four congruent sides. A square meets both requirements, so it satisfies the definition of a rectangle. A rectangle only meets the first, so a $4$ by $9$ rectangle is a rectangle that is not a square.
95. Counterexample: a kite with sides $5$, $5$, $8$, $8$ in that order around the figure. It has two pairs of congruent sides, but the congruent sides are adjacent rather than opposite, and no pair of its sides is parallel, so it is not a parallelogram. The student overlooked where the congruent sides are: a parallelogram requires the congruent pairs to be opposite each other, and requires the opposite sides to be parallel.
96. Watch how the diagonals cross and compare their lengths. In all three figures the diagonals bisect each other. If they are congruent but meet at a slant, the figure is a rectangle. If they meet at a right angle but have different lengths, it is a rhombus. If they are congruent and meet at a right angle, it is a square.

Workbook-only items

Page 2, property table.

Page 4, symmetry sketch table. Square: $4$ lines — both midlines and both diagonals. Rectangle that is not a square: $2$ lines — the midlines only. Rhombus that is not a square: $2$ lines — the diagonals only. Parallelogram that is neither: $0$ lines. Isosceles trapezoid: $1$ line — the midline running between the two bases.

Page 6, complete each sentence. Every rectangle, rhombus, and square is a parallelogram. A square is both a rectangle and a rhombus. A parallelogram is not always a rectangle. A trapezoid has exactly one pair of parallel sides, so a parallelogram is not a trapezoid.

Page 10, fill-ins. One diagonal splits any quadrilateral into 2 triangles. Each triangle's angles add to 180°, so the quadrilateral's four angles add to 360°. In a parallelogram, opposite angles are congruent and consecutive angles are supplementary (they add to 180°).

Page 14, fill in the property. In a parallelogram, opposite sides are congruent. In a rhombus or square, all four sides are congruent. In a rectangle, the diagonals are congruent. In an isosceles trapezoid, the legs are congruent. In any parallelogram, the diagonals bisect each other.