Geometry Workbook — Chapter 10: Properties of Quadrilaterals

SOL G.PC.1 (a, c, d) · Companion to Textbook Chapter 10

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 132.

PAGE 1 — Chapter opener

Chapter 10 · Properties of Quadrilaterals

Standard G.PC.1 (a, c, d)

In this chapter you will:

• Name the six families the standard lists and give each one's defining condition
• State and prove the four properties of a parallelogram
• Say what a rectangle adds, what a rhombus adds, and why those two are different
• Say why a square needs no properties of its own
• Use the properties specific to a trapezoid and an isosceles trapezoid
• Verify a property with a construction instead of asserting it

Words to know: quadrilateral · parallelogram · rectangle · rhombus · square · trapezoid · isosceles trapezoid · base · leg · base angles · diagonal · bisect · perpendicular · consecutive angles · defining condition · inherited property · specific to · construction

Conventions: a trapezoid has exactly one pair of parallel sides, so no parallelogram is a trapezoid. Bisect each other ≠ congruent. An inherited property is never an answer to "what makes this a rhombus?" The arcs stay on the page.

PAGE 2 — The six families

10.1 One Defining Condition Each

FIGURE: fig1-the-quadrilateral-families.png (full width)

Fill in the table from the figure.

1. Give the defining condition of a rhombus. ______________________________

2. Give the defining condition of a trapezoid. ______________________________

3. Which two families is a square in at once? ______________ and ______________

PAGE 3 — The hierarchy

Where Each Family Sits

FIGURE: fig2-the-hierarchy.png (full width)

4. Why do the trapezoid box and the parallelogram box not touch?

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5. Which region holds the squares, and why is it where it is?

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6. In one sentence each: a family's definition is ____________________ , and a theorem is ____________________ .

Downward, every property is inherited. Upward, none is. That asymmetry is the source of most errors in this chapter.

PAGE 4 — Practice · naming from conditions

Practice

7. Both pairs of opposite sides parallel, one right angle. Name it. ______________

8. Exactly one pair of parallel sides. Can it be a parallelogram? ______ Why? ____________________

9. Is every square a rhombus? ______ Explain. ____________________

10. Is every rhombus a square? ______ Explain. ____________________

11. Is every rectangle a parallelogram? ______ Explain. ____________________

12. Is every parallelogram a rectangle? ______ Explain. ____________________

PAGE 5 — Practice · most specific name

Name the Family

13. Four congruent sides and four right angles. Name every family it belongs to.

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14. Exactly one pair of parallel sides, congruent legs. ______________

15. Four congruent sides, no right angle. Most specific family: ______________

16. Can a trapezoid have two pairs of parallel sides in this course? ______ Why? ____________________

17. Application. A window frame has both pairs of opposite sides parallel and all four corners square. Most specific family: ______________ Which two conditions were checked?

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PAGE 6 — Error analysis and reasoning · 10.1

Think It Through

18. Error analysis. A student says a square is not a rectangle "because it has equal sides." Correct them.

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19. Reasoning. Why is a property true of all parallelograms automatically true of all rhombi, but not the other way round?

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20. Reasoning. Why does the exclusive definition of a trapezoid mean the two boxes cannot overlap?

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Exit ticket 10.1

21. Defining condition of a rectangle: ______________________________

22. Name the six families: ______________________________________________

PAGE 7 — A parallelogram's four properties

10.2 Four Properties, All Theorems

FIGURE: fig3-parallelogram-properties.png (full width)

23. Name the four properties.

    · ____________________ · ____________________

    · ____________________ · ____________________

24. Which one is the definition? ____________________ Which are theorems? ____________________

25. Where do the diagonals meet? ______ What is true of that point? ____________________

PAGE 8 — Proving opposite sides congruent

One Diagonal, Two Triangles

FIGURE: fig4-proving-opposite-sides-congruent.png (full width)

26. Name the two pairs of alternate interior angles and the reason.

    ____________________ and ____________________ , because ____________________

27. Name the criterion and the reason that give the opposite sides.

    Criterion: ____________ Reason for the shared side: ____________ Then: ____________

PAGE 9 — Proving the diagonals bisect

The Same Move Again

FIGURE: fig5-diagonals-bisect-each-other.png (full width)

28. Explain the difference between bisect each other and congruent.

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Watch this one. Reading bisect each other as congruent is the most common error in the chapter, and Lesson 10.3 is where it costs you.

PAGE 10 — Practice · parallelogram sides and angles

Practice

Parallelogram $ABCD$ throughout.

29. $AB = 9$, $BC = 14$. $CD =$ ______ $DA =$ ______

30. $m\angle A = 72°$. $m\angle B =$ ______ $m\angle C =$ ______ $m\angle D =$ ______

31. $m\angle B = 115°$. $m\angle D =$ ______ $m\angle C =$ ______

32. Diagonals meet at $M$, $AM = 7$. $AC =$ ______

33. Diagonals meet at $M$, $BD = 22$. $BM =$ ______

PAGE 11 — Practice · parallelograms with algebra

Solve for the Variable

34. $AB = 3x + 4$, $CD = 5x - 6$. $x =$ ______ $AB =$ ______

35. $BC = 2y + 7$, $AD = 4y - 9$. $y =$ ______ $BC =$ ______

36. $m\angle A = (2y + 10)°$, $m\angle B = (3y)°$. $y =$ ______ $m\angle A =$ ______ $m\angle B =$ ______

37. $m\angle A = (4z - 20)°$, $m\angle C = (2z + 30)°$. $z =$ ______ $m\angle A =$ ______

38. Diagonals meet at $M$, $AM = 2x + 1$, $MC = 4x - 7$. $x =$ ______ $AC =$ ______

39. Perimeter $= 46$ and $AB = 8$. $BC =$ ______

Before you start: write down which property you are about to use. Half the errors on this page are the right algebra on the wrong property.

PAGE 12 — Application, error analysis, and proof · 10.2

Think It Through

40. Application. A gate is a parallelogram $6$ ft wide and $4$ ft tall with a diagonal brace, meeting the other diagonal at $M$. One diagonal is $7.2$ ft. How far is $M$ from each of its endpoints?

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41. Error analysis. A student says a parallelogram's diagonals are congruent because they bisect each other. Correct them.

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42. Reasoning. Why does proving opposite sides congruent also prove opposite angles congruent, with no extra work?

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43. Reasoning. Why does one angle of a parallelogram determine all four?

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44. Write a two-column proof that the diagonals of a parallelogram bisect each other.

    Statement	Reason
    1.	1.
    2.	2.
    3.	3.
    4.	4.
    5.	5.
    6.	6.
    7.	7.

Exit ticket 10.2

45. $m\angle C = 58°$. $m\angle A =$ ______ $m\angle B =$ ______

46. Diagonals meet at $M$, $AC = 18$. $AM =$ ______

PAGE 13 — What a rectangle adds

10.3 Congruent Diagonals

FIGURE: fig6-rectangle-adds-congruent-diagonals.png (full width)

47. Name the two properties specific to a rectangle.

    · ____________________ · ____________________

48. Name one property it has that is inherited rather than specific. ____________________

PAGE 14 — What a rhombus adds

Perpendicular Diagonals

FIGURE: fig7-rhombus-adds-perpendicular-diagonals.png (full width)

49. Name the three properties specific to a rhombus.

    · ____________________ · ____________________ · ____________________

50. On that figure, are the diagonals congruent? ______ Explain. ____________________

51. The one-line rule that keeps the two apart:

    Rectangle → ____________________ . Rhombus → ____________________ .

52. Why do both still have diagonals that bisect each other? ____________________

PAGE 15 — Practice · rectangles and rhombi

Practice

53. Rectangle $ABCD$, $AC = 26$. $BD =$ ______ $AM =$ ______

54. Rectangle $ABCD$, $BD = 15$. $AC =$ ______ $MC =$ ______

55. Rhombus $ABCD$, diagonals meet at $M$. $m\angle AMB =$ ______

56. Rhombus $ABCD$, $m\angle ABC = 118°$. $m\angle ABD =$ ______

57. Rhombus $ABCD$, $m\angle BAD = 64°$. $m\angle BAC =$ ______ $m\angle ABC =$ ______

58. Rectangle $ABCD$, $AC = 4x + 3$, $BD = 6x - 11$. $x =$ ______ $AC =$ ______

59. Rectangle $ABCD$, $AM = 2y - 1$, $MC = y + 6$. $y =$ ______ $AC =$ ______

60. Rhombus $ABCD$, $AB = 5w - 2$, $BC = 3w + 8$. $w =$ ______ Perimeter $=$ ______

PAGE 16 — Practice · which family, and diagonal lengths

Name It, Then Measure It

61. A parallelogram with perpendicular diagonals. Most specific family: ______________

62. A parallelogram with congruent diagonals. Most specific family: ______________

63. A parallelogram with diagonals both congruent and perpendicular: ______________

64. Rhombus with diagonals $16$ and $12$. Side length $=$ ______

    Sketch the four right triangles the diagonals make. Which two lengths are the legs?

65. Rectangle with $AB = 9$ and $BC = 12$. Each diagonal $=$ ______

PAGE 17 — Application, error analysis, and reasoning · 10.3

Think It Through

66. Application. A carpenter checks a doorframe by measuring both diagonals and finds them equal. What has that verified? ____________________

    What has it not verified? ____________________

67. Error analysis. A student says a rhombus's diagonals are congruent because they bisect each other. Identify both errors.

    The conclusion: ____________________ The reason: ____________________

68. Reasoning. Explain why a rhombus's diagonals bisect its angles, using the congruent triangles they create.

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Exit ticket 10.3

69. Property specific to a rectangle's diagonals: ____________ To a rhombus's: ____________

70. Rhombus with diagonals $10$ and $24$. Side length $=$ ______

PAGE 18 — A square has no properties of its own

10.4 Both Families at Once

FIGURE: fig8-square-is-both.png (full width)

71. List the three diagonal properties a square has, and name the family each comes from.

    · ____________________ — from ____________ · ____________________ — from ____________ · ____________________ — from ____________

72. Why does a square need no property of its own? ____________________

PAGE 19 — The property table

Read Down, Then Read Across

FIGURE: fig9-specific-to-table.png (full width)

73. Read down the rhombus column. List what a rhombus has.

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74. Read across the "diagonals ≅" row. Name every family with a check. ____________________

75. Name a property that distinguishes nothing among parallelograms. ____________________

76. In one sentence, what does specific to mean? ____________________

A property with a check in every parallelogram column is inherited. It is a true statement and a useless answer.

PAGE 20 — Practice · specific versus inherited

Which One Identifies It?

77. Specific to a rhombus: diagonals bisect each other / diagonals perpendicular / opposite angles congruent. ____________________

78. Specific to a rectangle: opposite sides congruent / diagonals congruent / diagonals bisect each other. ____________________

79. A parallelogram with four congruent sides and one right angle. ______________

80. A quadrilateral has congruent diagonals. Must it be a rectangle? ______ Explain.

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PAGE 21 — Practice · most specific family

Name the Smallest Family That Fits

81. A parallelogram whose diagonals are perpendicular. ______________

82. A parallelogram whose diagonals are congruent. ______________

83. Four congruent sides and four right angles. ______________

84. Exactly one pair of parallel sides and congruent diagonals. ______________

85. Which families have all four sides congruent? ______________

86. Which families have diagonals that bisect each other? ______________

PAGE 22 — Application, error analysis, and reasoning · 10.4

Think It Through

87. Application. A tile has four congruent sides and perpendicular diagonals. Which family is confirmed? ______________

    What would still need checking to call it a square? ____________________

88. Error analysis. A student says a figure is a square because its diagonals bisect each other. What does that property establish, and what does it not?

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89. Reasoning. Why does a square have no properties that a rectangle and a rhombus do not already have between them?

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90. Reasoning. Why is "opposite sides are congruent" a poor answer to "what makes this a rectangle?"

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PAGE 23 — Exactly one pair

10.5 Trapezoid, and Isosceles Trapezoid

FIGURE: fig10-trapezoid-and-isosceles-trapezoid.png (full width)

91. What makes both of them trapezoids? ____________________

92. What makes the right-hand one isosceles? ____________________

93. Give the two leg lengths of the plain trapezoid: $AD =$ ______ $BC =$ ______

    Why is it not isosceles? ____________________

Label the bases and the legs on each figure before answering.

PAGE 24 — What congruent legs buy

Three Properties

FIGURE: fig11-isosceles-trapezoid-properties.png (full width)

94. Name the three properties.

    · ____________________ · ____________________ · ____________________

95. Are the diagonals congruent? ______ Do they bisect each other? ______

96. Why is an isosceles trapezoid not a parallelogram? ____________________

It shares one property with a rectangle — congruent diagonals — and not the other. Bisecting belongs to parallelograms, and this figure is not one.

PAGE 25 — Practice · trapezoid angles and diagonals

Practice

Isosceles trapezoid $ABCD$ with $\overline{AB} \parallel \overline{DC}$.

97. $m\angle A = 68°$. $m\angle B =$ ______

98. Same figure. $m\angle D =$ ______ $m\angle C =$ ______

99. $m\angle D = 105°$. Give the other three: ______ ______ ______

100. $AC = 17$. $BD =$ ______

101. $AD = 3x - 5$, $BC = x + 7$. $x =$ ______ Leg $=$ ______

102. $AC = 5y + 2$, $BD = 8y - 13$. $y =$ ______ $AC =$ ______

PAGE 26 — Practice · what does and does not follow

Careful Here

103. Diagonals meet at $M$. Is $AM = MC$? ______ Explain. ____________________

104. A trapezoid has legs of $7$ and $9$. Is it isosceles? ______ What follows about its base angles?

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105. Exactly one pair of parallel sides and congruent diagonals. Name it. ______________

106. Name every family in this chapter whose diagonals are congruent. ______________

PAGE 27 — Application, error analysis, and reasoning · 10.5

Think It Through

107. Application. A bridge truss panel is an isosceles trapezoid with bases $12$ ft and $8$ ft and legs of $5$ ft each. Perimeter $=$ ______

     Give the second leg's length without measuring it, and say how you know. ____________________

108. Error analysis. A student says an isosceles trapezoid's diagonals bisect each other because they are congruent. Correct them.

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109. Reasoning. Why are the base angles at one base congruent, and why are the angles at the other base supplementary to them?

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110. Reasoning. An isosceles trapezoid and a rectangle both have congruent diagonals. What else would you check to tell them apart?

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Exit ticket 10.5

111. $m\angle A = 74°$. $m\angle B =$ ______ $m\angle D =$ ______

112. $BD = 21$. $AC =$ ______ Is $AM = MC$? ______

PAGE 28 — The five constructions

10.6 What Each One Verifies

FIGURE: fig12-constructions-that-verify.png (full width)

113. Which construction verifies that two sides are parallel? ____________________

114. Which verifies that a diagonal bisects an angle? ____________________ Which family does that point to? ____________

115. Which verifies a right angle? ____________________

PAGE 29 — Verifying a rhombus

One Compass Opening

FIGURE: fig13-verify-a-rhombus-by-construction.png (full width)

116. What was the compass set to? ____________ From which points was it swung? ____________

117. What does it prove if all four vertices lie on the arcs? ____________________

118. Why must the arcs be left on the page? ____________________

A ruler says two sides look equal. A compass carries one length onto the other and shows whether they coincide. That is why the standard asks for constructions.

PAGE 30 — Practice · choosing the construction

Pick the Right Tool

119. Verifies a parallelogram: ____________________ Used how many times? ______

120. Verifies four congruent sides: ____________________

121. Verifies a right angle at a vertex: ____________________

122. Verifies that a diagonal bisects an angle: ____________________

123. Describe, in order, how to verify a rhombus using one compass setting.

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PAGE 31 — Practice · describing the check

Write the Procedure

124. How would you verify that a trapezoid is isosceles using a construction?

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125. How would you verify that a parallelogram is a rectangle using a construction?

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126. Opposite sides verified parallel, one angle verified right. Family established: ______________

PAGE 32 — Application, error analysis, and reasoning · 10.6

Think It Through

127. Application. A shop must confirm a metal panel is a true rhombus before welding. Describe the construction check, and say what is left on the panel as evidence.

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128. Error analysis. A student verifies that a parallelogram's diagonals bisect each other and concludes it is a rhombus. Explain the error, and name a construction that would settle it.

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129. Reasoning. What is the difference between measuring a property with a ruler and verifying it with a construction?

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130. Reasoning. Why is verifying four congruent sides enough to conclude the diagonals are perpendicular, without constructing the diagonals at all?

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Exit ticket 10.6

131. Name the five constructions G.PC.1d lists: ______________________________________________

132. Which verifies a rhombus? ____________ What stays on the page? ____________

PAGE 33 — Blank frames

Your Turn

FIGURE: fig14-blank-quadrilateral-frames.png (full width)

Mark the given parts first. Then decide which family the figure is in. Only then reach for a property.

A checklist for every problem in this chapter:

1. What am I told — sides, angles, or diagonals?
2. Which family does that put me in?
3. Is the property I want specific to that family, or inherited? (Either is fine to use; only a specific one identifies.)
4. If diagonals are involved: bisect or congruent or perpendicular? They are three different claims.

PAGE 34 — Chapter review

Chapter 10 Review

Review 1 (G.PC.1 a, c). In parallelogram $ABCD$, $m\angle A = (3x + 15)°$, $m\angle B = (5x + 5)°$, and the diagonals meet at $M$ with $AM = 2y - 3$ and $MC = y + 4$.

• Find $x$, then give all four angle measures.
• Find $y$, then give $AM$ and $AC$.
• Name the property used in each computation, and say which is a definition and which is a theorem.
• A different parallelogram has $AC = BD$. Name the most specific family it must be in, say what would still be needed to call it a square, and explain why $ABCD$ above cannot be in that family.

PAGE 35 — Chapter review, continued

Review 2 (G.PC.1a). For each description, name the most specific family, and name one property that is specific to it rather than inherited.

Review 3 (G.PC.1 a, c, d). A fabricator is checking panels. Each is meant to be an isosceles trapezoid with bases $14$ in and $10$ in and legs of $6$ in.

• Give the perimeter of a correct panel.
• One panel has $m\angle A = 71°$ at one end of the long base. Give the other three angles of a correct panel, with a reason for each.
• Describe the construction that verifies the legs are congruent, and say what is left on the panel as evidence.
• A panel comes back with congruent diagonals and diagonals that bisect each other. Explain why it cannot be the intended isosceles trapezoid, and name what it is instead.

PAGE 36 — Vocabulary check

Words to Know

quadrilateral · parallelogram · rectangle · rhombus · square · trapezoid · isosceles trapezoid · base · leg · base angles · diagonal · bisect · perpendicular · consecutive angles · defining condition · inherited property · specific to · construction

The three that carry the chapter:

• Defining condition — the one thing you check to decide whether a figure is in a family. Everything else is a theorem.
• Specific to — true of this family and not of the family it sits inside. Only these identify a figure.
• Bisect each other — each diagonal is cut in half at their intersection. Not the same as congruent, which is about the two whole diagonals.

Answer keys for every item are in Appendix A.