10-7 Quadrilaterals - Macmillan/McGraw-Hillmhschool.com/math/mathconnects/wa/assets/docs/558_563_wa...558-563 C10_L07_105853.indd 558 7/20/09 12:11:40 AM Lesson 10-7 Quadrilaterals

558 Chapter 10 Geometry: Angles and Polygons

10-7 Quadrilaterals

MAIN IDEAI will classify quadrilaterals and find missing angle measures in quadrilaterals.

New Vocabulary quadrilateral

rectangle

square

parallelogram

rhombus

trapezoid

Hands-On Mini Lab The figure below is a quadrilateral , since it has four sides and four angles.

Step 1 Draw a quadrilateral.

Step 2 Pick one vertex and draw a segment to the opposite vertex.

1. How many triangles were formed? 2

2. Use the relationship among the angle measures in a triangle to find the sum of the angle measures in a quadrilateral. Explain. 360°; 2(180°) = 360°

The angles of a quadrilateral have a special relationship.

KEY CONCEPT Angles of a Quadrilateral

Words The sum of the measures of the angles of a quadrilateral is 360°.

Model w°x°z°

y°

Symbols w° + x° + y° + z° = 360°

EXAMPLE Find Angle Measures

1 Algebra Find the value of x in the 65°

x°

85°

quadrilateral.

Since the sum of the angle measures in a quadrilateral is 360°, x + 65 + 85 + 90 = 360.

x + 65 + 85 + 90 = 360 Write the equation.x + 240 = 360 Add 65, 85, and 90.

- 240 = - 240 Subtract 240 from each side. x = 120 Simplify.

So, the value of x is 120.

558-563 C10_L07_105853.indd 558 7/20/09 12:11:40 AM

Lesson 10-7 Quadrilaterals 559

Real-World EXAMPLE Classify Quadrilaterals

2 Classify the quadrilaterals labeled 1 and 2 in the quilt piece.

Figure 1 is a square. Figure 2 is a rhombus.

Reading MathCongruent Angles The red arcs show congruent angles.

1 2

KEY CONCEPTQuadrilateral Figure Characteristics

Rectangle

• Opposite sides congruent

• All angles are right angles

• Opposite sides parallel

Square

• All sides congruent

• All angles are right angles

• Opposite sides parallel

Parallelogram

• Opposite sides congruent

• Opposite sides parallel

• Opposite angles congruent

Rhombus

• All sides congruent

• Opposite sides parallel

• Opposite angles congruent

Trapezoid

• Exactly one pair of opposite sides parallel

Classifying Quadrilaterals

Extra Examples at ca.gr5math.com

558-563 C10_L07_105853.indd 559 7/20/09 12:11:46 AM

560 Chapter 10 Geometry: Angles and Polygons

EXAMPLE Find Angle Measures

3 What is the value of x in the

parallelogram at the right?

You know that in a parallelogram, opposite angles are congruent. Since the angle opposite the missing measure has a measure of 70°, x = 70.

Check 70° + 110° + 70° + 110° = 360° �

Personal Tutor at ca.gr5math.com

Algebra Find the value of x in each quadrilateral. See Example 1 (p. 558)

1. 105°

120°x°

60° 75 2. 107°

63°58°

x° 132

3.

x°40°

146° 116° 58 4. 19°x °

161

5. Classify each quadrilateral. See Example 2 (p. 559)

C13-12P-829631-A.ai

C13-13P-829631-A.ai 6. Safari Stereo created the logo shown below for use in their new advertisement. Classify the quadrilaterals used in the logo. See Example 2 (p. 559) trapezoids and parallelograms

7. Algebra Find the value of x in the parallelogram at the right.

65°

115° 65°

x°See Example 3 (p. 560) 115

8. Describe two different real-world items that are shaped like quadrilaterals. Then classify those quadrilaterals. Sample answer: chalkboard–quadrilateral or rectangle; floor tile–square

70°

x°

110°

110°

The first figure is a rectangle. The second figure is a square.

558-563 C10_L07_105853.indd 560 7/20/09 12:11:51 AM

Lesson 10-7 Quadrilaterals 561

Algebra Find the value of x in each quadrilateral. See Examples 1, 3 (pp. 558, 560)

9.

x° 65°

120°80° 95 10.

110° 110°

70°x° 70 11.

105°98°

x° 67

12.

110°x°

55°95° 100 13.

105°

x°

60°

60° 135 14. 75°

85°x°

115° 85

Classify each quadrilateral. See Example 2 (p. 559)

15. rectangle 16. square 17.

18. trapezoid 19. 20. rhombus

21. Many aircraft display the shape of the American flag slightly distorted to indicate motion. Classify each quadrilateral. See Example 2 (p. 559)

22. Classify each quadrilateral. See Example 2 (p. 559)

Measure the angles and sides of each quadrilateral. Then use this information to classify each quadrilateral.

23. all 90°; 24. 60°; 60°; 25. 2.3 cm,

1.3 cm, 2.3 cm, 1.3 cm; rectangle

120°; 120°; all 1.7 cm;

rhombus

parallelogram

quadrilateral

PRACTICEEXTRASee page 684.

The first figure is a rectangle. The second figure is a trapezoid.

The first figure is a rectangle. The second figure is a parallelogram.

100°; 90°; 75°; 95°; 2.1 cm, 2.5 cm, 2.5 cm, 2 cm; quadrilateral

558-563 C10_L07_105853.indd 561 7/20/09 12:11:59 AM

562 Chapter 10 Geometry: Angles and Polygons

26. Grace sorted a set of quadrilaterals into two categories according to a certain rule. The shapes that followed the rule were put in Set A, and the shapes that did not follow the rule were put in Set B.

Set Asquare rhombus

Set Brectangle parallelogram trapezoid

What rule did Grace use to sort the quadrilaterals? Sample answer: The shapes have all sides congruent. 27. Use the Internet or another source to look up the meaning of the

term isosceles trapezoid. Explain how an isosceles trapezoid is related to an isosceles triangle. Then draw an example of an isosceles trapezoid. See margin.

28. OPEN ENDED Draw a rectangle using a ruler and protractor. Label each side and angle with its measure. Label all parallel and perpendicular lines. See margin.

29. NUMBER SENSE Three of the angle measures of a quadrilateral are congruent. Without calculating, determine if the measure of the fourth angle in each of the following situations is greater than, less than, or equal to 90°. Explain your reasoning. See margin.a. The three congruent angles each measure 89°.

b. The three congruent angles each measure 90°.

c. The three congruent angles each measure 91°.

CHALLENGE Determine whether each statement is sometimes, always, or never true. Explain your reasoning. 30–33. See margin.

30. A rhombus is a square. 31. A quadrilateral is a parallelogram.

32. A rectangle is a square. 33. A square is a rectangle.

34. SELECT A TOOL Quentin is designing a logo for his lawn care company. He wants to use polygons in the logo. Which of the following tools might Quentin use to design several sample logos? Justify your selection(s). Use the tool(s) to create a sample logo using polygons. Then classify each polygon you used in the sample logo. See margin.

real objects paper/pencil technology

558-563 C10_L07_105853.indd 562 7/20/09 12:12:02 AM

Lesson 10-7 Quadrilaterals 563

35. Make a diagram that shows the relationship between each of the following shapes: rectangle, parallelogram, square, rhombus, quadrilateral, and trapezoid. Then write a few sentences that explain your diagram. See margin.

36 The drawing below shows the shape of Hinto’s patio. B

Hinto’s PatioA

75°

Find the measure of ∠A.

A 75°

B 105°

C 165°

D 195°

37 A parallelogram is shown below. Find the measure of ∠M to the nearest degree. H

F 30° H 120°

G 60° J 150°

Algebra Find the value of x in each triangle. (Lesson 10-6)

38. 105 39. 40 40. 74

41. Tamika is working on a group project with 3 other students. They must decide who will be the researcher, the artist, the writer, and the presenter. How many ways can they choose to assign the jobs? (Lesson 10-5) 24 ways

42. Geometry Determine if the pair of lines marked on the polygon is best described as parallel, perpendicular, or neither. (Lesson 10-4) parallel

43. Mallory needs 4 3 _ 4 cups of flour for a bread recipe. Write 4 3 _ 4

as an improper fraction. (Lesson 4-4) 19 _ 4

Self-Check Quiz at ca.gr5math.com

36 The drawing below shows the shape of Hinto’s patio. B

Hinto’s PatioA

75°

Find the measure of ∠A.

A 75°

B 105°

C 165°

D 195°

37 A parallelogram is shown below. Find the measure of ∠M to the nearest degree. H

F 30° H 120°

G 60° J 150°

558-563 C10_L07_105853.indd 563 7/20/09 12:12:08 AM