Download Quadrilaterals

10-6
MAIN IDEA
Identify and classify
quadrilaterals.
New Vocabulary
quadrilateral
parallelogram
trapezoid
rhombus
Math Online
glencoe.com
• Extra Examples
• Personal Tutor
• Self-Check Quiz
Quadrilaterals
VIDEO GAMES The general
shape of a video game
controller is shown.
1. Describe the angles inside
the four-sided figure.
2. Which sides of the figure
appear to be parallel?
3. Which sides appear to be
congruent?
NC-03A-880486.ai
A quadrilateral is a closed figure with four sides and four
angles.
Quadrilaterals are named based on their sides and angles. The diagram
be
shows how quadrilaterals are related. Notice how it goes from the most
general to the most specific.
Quadrilateral
Parallelogram
Trapezoid
quadrilateral with both pairs
of opposite sides parallel
and congruent
quadrilateral with exactly
one pair of parallel sides
Rectangle
parallelogram with
4 right angles
Rhombus
parallelogram with
4 congruent sides
parallelogram with
4 right angles
and 4 congruent sides
Square
The name that best describes a quadrilateral is the one that is most
specific.
C10-134C-829633
• If a quadrilateral has
all the properties of a parallelogram and a
rhombus, then the MA
best description of the quadrilateral is a rhombus.
Parallel Lines The sides
with matching arrows are
parallel.
• If a quadrilateral has all the properties of a parallelogram, rhombus,
rectangle, and square, then the best description of the quadrilateral
FONT CONVERSION
is a square.
CH 10
TECH
ART FILE: C10-134C-829633
Glencoe 10-6 Quadrilaterals 10555
Lesson
533
CUSTOMER:
JOB NUMBER:
CONVERTED FONT:
CS
DATE:
7-30-07
Draw and Classify Quadrilaterals
Draw a quadrilateral that satisfies each set of conditions. Then
classify each quadrilateral with the name that best describes it.
1 a parallellogram with four right angles and four congruent sides
Draw one right angle. The two segments
should be congruent.
Draw a second right angle that shares
one of the congruent segments. The third
segment drawn should be congruent to
the first two segments drawn.
Check for Reasonableness
Use a ruler and a
protractor to measure
the sides and angles to
verify that your drawing
satisfies the given
conditions.
Connect the fourth side of the quadrilateral.
All four angles should be right angles, and all
four sides should be congruent.
The figure is a square.
2 a quadrilateral with opposite sides parallel
Draw two parallel sides of equal length.
Connect the endpoints of these
two sides so that two new parallel
sides are drawn.
The figure is a parallelogram.
a. Sample answer: trapezoid
Draw a quadrilateral that satisfies each set of conditions. Then
classify each quadrilateral with the name that best describes it.
a. a quadrilateral with exactly one pair of parallel sides
b. a parallelogram with four congruent sides
A quadrilateral can be separated into two triangles,
A and B. Since the sum of the angle measures of
b. Sample answer: rhombus
each triangle is 180°, the sum of the angle measures
of the quadrilateral is 2 · 180, or 360°.
B
A
Key Concept
Angles of a Quadrilateral
534
Words
The sum of the measures of the
angles of a quadrilateral is 360°.
Algebra
w + x + y + z = 360
Chapter 10 Geometry: Polygons
Model
w°
x°
z°
y°
HOMEWORK
For
Exercises
9–14,
23–24
15–22
HELP
See
Examples
Classify each quadrilateral with the name that best describes it.
9.
10.
1, 2
11.
quadrilateral
trapezoid
square
3
12.
13.
Exercise Levels
A: 9–24
B: 25–37
C: 38–45
14.
trapezoid
rhombus
parallelogram
ALGEBRA Find the missing angle measure in each quadrilateral.
15.
16.
106°
x°
131°
18.
24. 65°; 73°; 17°;
138°; Sample
answer: The sum
of the measures of
∠b and ∠c is 180°
- 90° or 90° since
these angles are
complementary.
The measure of
∠a is 360° - 115°
- 90° or 65° since
these angles form
a quadrilateral.
The measure of
∠b is 180° - 42°
- 65° or 73° since
these angles form
a triangle. The
measure of ∠c is
90° - 73° or 17°
since ∠b and ∠c
are complementary.
The measure of
∠d is 180° - 42°
or 138° since these
angles are
supplementary.
536
55°
x°
143°
x°
67°
17.
80°
89°
72°
56°
113°
67°
119°
19.
32°
x°
130°
x°
20.
x°
100°
100°
80°
90°
80°
21. ALGEBRA Find m∠B in quadrilateral ABCD if m∠A = 87°, m∠C = 135°,
and m∠D = 22°. 116°
22. ALGEBRA What is m∠X in quadrilateral WXYZ if m∠W = 45°, m∠Y = 128°,
and ∠Z is a right angle? 97°
23. LANDSCAPE Identify the shapes of the
B
bricks used in the design at the right.
Use the name that best describes
each brick.
Bricks A, B, and D are rectangles and
brick C is a square.
24. MEASUREMENT Find each of the
missing angle measures a, b, c,
and d in the figure at the right.
Justify your answers.
A
C
D
a
42°
b
d
115°
c
Find the missing measure in each quadrilateral with the given angle
measures.
25. 37.5°, 78°, 115.4°, x°
129.1°
27. 25.5°, x°, 165.9°, 36.8°
Chapter 10 Geometry: Polygons
131.8°
26. x°, 108.3°, 49.8°, 100°
101.9°
28. 79.1°, 120.8°, x°, 65.7°
94.4°
48. Which of the following is a correct
46. Identify the name that
drawing of a quadrilateral with all
sides congruent and with four right
angles? A
does not describe the
quadrilateral shown. D
A square
A
B rectangle
C rhombus
D trapezoid
B
47. Which statement is always true about
a rhombus? J
F It has 4 right angles.
G The sum of the measures of the
angles is 180°.
C
H It has exactly one pair of parallel
sides.
J
It has 4 congruent sides.
D
49. REASONING Neva, Sophie, and Seth have a turtle, a dog, and a hamster
for a pet, but not in that order. Sophie’s pet lives in a glass aquarium and
does not have fur. Neva never has to give her pet a bath. Who has what
pet? Use the logical reasoning strategy. (Lesson 10-5) Neva: hamster, Sophie: turtle, Seth: dog
Classify each triangle by its angles and by its sides.
50.
(Lesson 10-4)
51.
obtuse, isosceles
52.
right, scalene
acute, equilateral
53. LETTERS How many permutations are possible of the letters in the word
Fresno?
(Lesson 9-4)
720
Find the sales tax or discount to the nearest cent.
54. $54 jacket; 7% sales tax
$3.78
55. $23 hat; 15% discount
PREREQUISITE SKILL Solve each proportion.
56.
538
x
_3 = _
45
5
75
57.
18
_a = _
3
7
42
Chapter 10 Geometry: Polygons
(Lesson 7-7)
58.
$3.45
(Lesson 6-6)
28
_7 = _
36
9
m
59.
16
3.5
_
=_
7
t
32
60.
_3 = _c 2.5
6
5
Find a Missing Measure
73°
3 ALGEBRA Find the value of x
59°
in the quadrilateral shown.
C10-140C-829633
Words
The sum of the measures is 360°.
Variable
Let x represent the missing measure.
Equation
85 + 73 + 59 + x = 360
MA
85 + 73 + 59 + x = 360
217 + x = 360
- 217
= - 217
_______________________
x = 143
FONT CONVERSION
ART FILE: C10-140C-829633
CH 10
CUSTOMER:
JOB NUMB
Write the equation.
Simplify.
Glencoe
CS
CONVERTED FONT:
DATE:
Subtract 217 fromEDITED
each side.
BY:
DATE:
TIME:
So, the missing angle measure is 143°.
created@ NETS
REVISION:
1
simple
c. ALGEBRA Find the value of x in the
quadrilateral shown.
only altered
2
3
mod.
blackline
48° 157°
complex
greyscale
55°
x°
C10-145C-829633
Check for Reasonableness
Since ∠x is an obtuse
angle, m∠x should be
between 90° and 180°.
Since 90° < 143° < 180°,
the answer is reasonable.
x°
85°
Write and solve an equation.
Examples 1, 2
(p. 534)
Classify each quadrilateral with the name that best describes it.
2.
3.
C10-144C-829633
4. BOATS The photo shows a sailboat called
CS
CH 10
2
Example 3
a schooner. What type of quadrilateral
MA
does the indicated sail best represent?
5. ALGEBRA In quadrilateral DEFG,
(p. 535)
m∠D = 57°, m∠E = 78°, m∠G = 105°.
What is m∠F?
ALGEBRA Find the missing angle measure in each quadrilateral.
NC-04A-880486.ai
6.
146° x°
55°
59°
TECH
10555
7-26-07
JOB NUMBER:
DATE:
DATE:
TIME:
only altered@ NETS
FONT CONVERSION
Glencoe
ART FILE: C10-143C-829633
CUSTOMER:
CONVERTED FONT:
EDITED BY:
created@ NETS
$$
."
1.
MA
C11-061A-874046
7.
108° 73°
115°
x°
8.
71°
164°
be
x°
C10-148C-829633
C10-146C-829633
MA
C10-147C-829633
MA
Lesson 10-6
Quadrilaterals 535
MA
ART For Exercises 29–31, identify the types of triangles and quadrilaterals used
in each quilt block pattern. Use the names that best describe the figures.
29.
30.
#!
26
DRAWING
QUADRILATERALS
31.
#!
26
Determine
whether
#!
26 described below
each figure
can be drawn. If the figure can be drawn, draw it. If not, explain why not.
32. a quadrilateral that is both a rhombus and a rectangle
33. a trapezoid with three right angles
34. a trapezoid with two congruent sides
ALGEBRA Find the value of x in each quadrilateral.
EXTRA
PRACTICE
35.
See pages 694, 713.
H.O.T. Problems
2x°
2x°
2x°
2x°
36.
135°
37.
135°
x°
x°
100°
x°
x°
100°
CHALLENGE For Exercises 38 and 39, refer to the
Parallelogram
Properties
table that gives the properties of several
1
A, C
C11-067A-874046
C11-066A-874046
parallelograms.
Property A states
that both pairs
2
A, B, C
C11-068A-874046
of opposite sides are parallel and congruent.
3
38. If property C states that all four sides are
A, B
congruent, classify parallelograms 1–3. Justify your response.
39. If parallelogram 3 is a rectangle, describe Property B. Justify your response.
REASONING Determine whether each statement is sometimes, always, or
never true. Explain your reasoning.
40. A quadrilateral is a trapezoid.
41. A trapezoid is a parallelogram.
42. A square is a rectangle.
43. A rhombus is a square.
44. FIND THE ERROR Isabelle and John are describing a rectangle. Who is more
accurate? Explain.
a quadrilateral with 2
pairs of parallel sides
a parallelogram with
4 right angles
Isabelle
45.
John
WR ITING IN MATH The diagonals of a rectangle are congruent, and the
diagonals of a rhombus are perpendicular. Based on this information, what
can you conclude about the diagonals of a square? of a parallelogram?
Explain your reasoning.
Lesson 10-6 Quadrilaterals
537