10-7
Quadrilaterals
Hands-On Mini Lab
MAIN IDEA
I will classify
quadrilaterals
and find missing
angle measures in
quadrilaterals.
New Vocabulary
quadrilateral
rectangle
square
parallelogram
rhombus
trapezoid
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 $$
quadrilateral.
Explain. 360°; 2(180°) = 360°
The angles of a quadrilateral have a special relationship.
KEY CONCEPT
Words
Model
Angles of a Quadrilateral
The sum of the measures of the angles of a
quadrilateral is 360°.
z°
Symbols w° + x° + y° + z° = 360°
w°
x°
y°
$"
Find Angle Measures
EXAMPLE
1 Algebra Find the value of x in the
quadrilateral.
Since the sum of the angle measures in a
quadrilateral is 360°, x + 65 + 85 + 90 = 360.
x + 65 + 85 + 90
x + 240
- 240
x
= 360
= 360
= - 240
= 120
85°
65°
x°
Write the equation.
Add 65, 85, and
90.
$"
Subtract 240 from each side.
Simplify.
So, the value of x is 120.
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Chapter 10 Geometry: Angles and Polygons
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KEY CONCEPT
Quadrilateral
Classifying Quadrilaterals
Figure
Characteristics
• Opposite sides
congruent
Reading Math
Congruent Angles
The red arcs show
congruent angles.
• All angles are right
angles
Rectangle
• Opposite sides
parallel
$$" • All sides congruent
• All angles are right
angles
Square
• Opposite sides
parallel
• Opposite sides
congruent
$$"
• Opposite sides
parallel
Parallelogram
• Opposite angles
congruent
$$" • All sides congruent
• Opposite sides
parallel
Rhombus
• Opposite angles
congruent
Trapezoid
• Exactly one pair of
$$"opposite sides
parallel
Real-World
EXAMPLE
Classify Quadrilaterals
$"
$"
2 Classify the quadrilaterals labeled
1 and 2 in the quilt piece.
Figure 1 is a square. Figure 2 is a
rhombus.
1
2
$"
37
Extra Examples at ca.gr5math.com
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Lesson 10-7 Quadrilaterals 559
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EXAMPLE
Find Angle Measures
3 What is the value of x in the
parallelogram at the right?
You know that in a parallelogram,
opposite angles are congruent. Since
the angle opposite the missing
measure has a measure of 70°, x = 70.
Check
x°
110°
110°
70°
70° + 110° + 70° + 110° = 360° $"
Personal Tutor at ca.gr5math.com
Algebra Find the value of x in each quadrilateral.
1.
2.
146°
x° 107°
58°
120°
x°
3.
75
60°
105°
See Example 1 (p. 558)
58
116°
4.
°
x
40°
$"
132
63°
x°
19°
161
$"
5. Classify each quadrilateral. See Example 2 (p. 559)
$"
The first figure is a rectangle. The
second
figure is a square.
$"
6. Safari Stereo created the logo shown below for use in their new
C13-13P-829631-A.ai
C13-12P-829631-A.ai
advertisement. Classify the quadrilaterals used in the logo.
See Example 2 (p. 559)
trapezoids and parallelograms
4
4
$""
7. Algebra Find the value of x in the parallelogram at the right.
See Example 3 (p. 560)
8.
115
115°
65°
65°
x°
Describe two different real-world items that are shaped like
quadrilaterals. Then classify those quadrilaterals. Sample answer:
chalkboard–quadrilateral or rectangle; floor tile–square$$
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Chapter 10 Geometry: Angles and Polygons
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EXTRA
%
)# E
# T4 IC
!C
2A
0R
P
See page 684.
Algebra Find the value of x in each quadrilateral.
9.
80°
10.
95°
x°
70°
110°
65°
x°
12.
95
120°
100
55°
13.
See Examples 1, 3 (pp. 558, 560)
70
11.
135
14.
x°
$$
$$
x°
105°
110°
67
98°105°
110°
60°
x°
75°
85
115°
$"
°
x
85°
60°
Classify each quadrilateral. See Example 2 (p. 559)
15.
rectangle
$"
16.
square
$"
17.
$"
parallelogram
18.
trapezoid 19.
$"
rhombus
20.
$"
$"
quadrilateral
21. Many aircraft display the shape of the American flag slightly distorted
to indicate motion. Classify each quadrilateral. See Example 2 (p. 559)
$"
$"
$"
The first figure is a
rectangle. The second
figure is a parallelogram.
22. Classify each quadrilateral. See Example 2 (p. 559)
The first figure is a
rectangle. The second
figure is a trapezoid.
$$"
$"
Measure the angles and
37 sides of each quadrilateral. Then use this
information to classify each quadrilateral.
23.
all 90°;
24.
2.3 cm,
1.3 cm,
2.3 cm,
1.3 cm;
rectangle
60°; 60°; 25.
120°; 120°;
all 1.7 cm;
rhombus
$"
$"
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100°; 90°;
75°; 95°;
2.1 cm,
2.5 cm,
2.5 cm, 2 cm;
quadrilateral
Lesson 10-7 Quadrilaterals 561
$"
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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 A
square
Set B
rhombus
$"
$"
rectangle
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
562
paper/pencil
technology
Chapter 10 Geometry: Angles and Polygons
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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 Patio
37 A parallelogram is shown below. Find
the measure of ∠M to the nearest
degree. H
A
.
A 75°
B 105° $""
F 30°
H 120°
C 165°
G 60°
J 150°
D 195°
Find the measure of ∠A.
$""
Algebra Find the value of x in each triangle.
38.
75°
35.
105
(Lesson 10-6)
40
39.
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
3
3
43. Mallory needs 4_ cups of flour for a bread recipe. Write 4_
4
4
19
as an improper fraction. (Lesson 4-4)
4
_
Self-Check Quiz at ca.gr5math.com
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