9.6 Quadrilaterals and Other Polygons

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87.HEIGHT OF A TREE A tree casts a shadow of

29 feet at the same time as a vertical yardstick casts a shadow of 2.5 feet. Find the height of the tree.

3 ft

2.5 ft

29 ft

88.GEOGRAPHY The diagram below shows how a laser beam was pointed over the top of a pole to the top of a mountain to determine the elevation of the mountain. Find h.

	Laser	beam
	Laser	h
		h

9-ft

pole

5 ft


20 ft				6,000 ft

a. What is m(CD)?

What is m(AD)?

767

89.FLIGHT PATH An airplane ascends 200 feet as it ﬂies a horizontal distance of 1,000 feet, as shown in the following ﬁgure. How much altitude is gained as it ﬂies a horizontal distance of 1 mile?

(Hint: 1 mile 5,280 feet.)

xft

200 ft

1,000 ft

1 mi

WRITING

90.Tell whether the statement is true or false. Explain your answer.

a.Congruent triangles are always similar.

b.Similar triangles are always congruent.

91.Explain why there is no SSA property for congruent triangles.

REVIEW

Find the LCM of the given numbers.

92.

16, 20

93.

21, 27

Find the GCF of the given numbers.

94.

18, 96

95.

63, 84

S E C T I O N 9.6

Quadrilaterals and Other Polygons

Recall from Section 9.3 that a polygon is a closed geometric ﬁgure with at least three line segments for its sides. In this section, we will focus on polygons with four sides, called quadrilaterals. One type of quadrilateral is the square. The game boards for Monopoly and Scrabble have a square shape. Another type of quadrilateral is the rectangle. Most picture frames and many mirrors are rectangular. Utility knife blades and swimming ﬁns have shapes that are examples of a third type of quadrilateral called a trapezoid.

Objectives

1Classify quadrilaterals.

2Use properties of rectangles to ﬁnd unknown angle measures and side lengths.

3Find unknown angle measures of trapezoids.

4Use the formula for the sum of the angle measures of a polygon.

1 Classify quadrilaterals.

A quadrilateral is a polygon with four sides. Some common quadrilaterals are shown below.

Parallelogram	Rectangle	Square	Rhombus	Trapezoid
(Opposite sides	(Parallelogram with	(Rectangle with	(Parallelogram with	(Exactly two
parallel)	four right angles)	sides of equal length)	sides of equal length)	sides parallel)

EXAMPLE 1

768	Chapter 9 An Introduction to Geometry

We can use the capital letters that label the vertices of a quadrilateral to name it. For example, when referring to the quadrilateral shown on the right, with vertices A, B, C, and D, we can use the notation quadrilateral

ABCD.

D C

A B

Quadrilateral ABCD

The Language of Mathematics When naming a quadrilateral (or any other polygon), we may begin with any vertex. Then we move around the ﬁgure in a clockwise (or counterclockwise) direction as we list the remaining vertices.

Some other ways of naming the quadrilateral above are quadrilateral ADCB, quadrilateral CDAB, and quadrilateral DABC. It would be unacceptable to name it as quadrilateral ACDB, because the vertices would not be listed in clockwise (or counterclockwise) order.

A segment that joins two nonconsecutive vertices of a polygon is called a diagonal of the polygon. Quadrilateral ABCD shown below has two diagonals, AC and BD.

D C

A B

2Use properties of rectangles to ﬁnd unknown angle measures and side lengths.

Recall that a rectangle is a quadrilateral with four right angles. The rectangle is probably the most common and recognizable of all geometric ﬁgures. For example, most doors and windows are rectangular in shape. The boundaries of soccer ﬁelds and basketball courts are rectangles. Even our paper currency, such as the $1, $5, and $20 bills, is in the shape of a rectangle. Rectangles have several important characteristics.

Properties of Rectangles

In any rectangle:

1.All four angles are right angles.

2.Opposite sides are parallel.

3.Opposite sides have equal length.

4.The diagonals have equal length.

5.The diagonals intersect at their midpoints.

In the ﬁgure, quadrilateral WXYZ is a rectangle. Find each measure:

a. m( YXW) b. m(XY) c. m(WY) d. m(XZ)

Strategy We will use properties of rectangles to ﬁnd the unknown angle measure and the unknown measures of the line segments.

WHY Quadrilateral WXYZ is a rectangle.

Z	8 in.		W
6 in.		5 in.
6 in.	A
	A
Y			X
Y			X

9.6 Quadrilaterals and Other Polygons

769

Solution

Self Check 1

a. In any rectangle, all four angles are right angles. Therefore, YXW is a right

In rectangle RSTU shown below,

angle, and m( YXW) 90°.

the length of RT is 13 ft. Find

and

are opposite sides of the rectangle, so they have equal length.

each measure:

Since the length of

is 8 inches, m(

) is also 8 inches.

a. m( SRU)

and

are diagonals of the rectangle, and they intersect at their

b. m(ST)

midpoints. That means that point A is the midpoint of WY. Since the length of

c. m(

)

WA is 5 inches, m(WY) is 2 5 inches, or 10 inches.

12 ft

d. The diagonals of a rectangle are of equal length. In part c, we found that the

d. m(SG)

length of

is 10 inches. Therefore, m(

) is also 10 inches.

We have seen that if a quadrilateral has four right angles, it is a rectangle. The

following statements establish some conditions that a parallelogram must meet to

5 ft

ensure that it is a rectangle.

Now Try Problem 27

Parallelograms That Are Rectangles

If a parallelogram has one right angle, then the parallelogram is a

rectangle.

If the diagonals of a parallelogram are congruent, then the parallelogram

is a rectangle.

EXAMPLE 2

Construction A carpenter wants to build a shed with a

9-foot-by-12-foot base. How can he make sure that the foundation has four right-

angle corners?

12 ft

9 ft

12 ft

Strategy The carpenter should ﬁnd the lengths of the diagonals of the

foundation.

WHY If the diagonals are congruent, then the foundation is rectangular in shape

and the corners are right angles.

Solution The four-sided foundation, which we will label as parallelogram ABCD,

has opposite sides of equal length. The carpenter can use a tape measure to ﬁnd the

lengths of the diagonals

and

. If these diagonals are of equal length, the

foundation will be a rectangle and have right angles at its four corners. This process

is commonly referred to as “squaring a foundation.” Picture framers use a similar

process to make sure their frames have four 90° corners.

Now Try Problem 59

3 Find unknown angle measures of trapezoids.

A trapezoid is a quadrilateral with exactly two sides parallel. For the trapezoid shown on the next page, the parallel sides AB and DC are called bases. To distinguish between the two bases, we will refer to AB as the upper base and DC as the lower base. The angles on either side of the upper base are called upper base angles, and the angles on either side of the lower base are called lower base angles. The nonparallel sides are called legs.

770	Chapter 9 An Introduction to Geometry

Self Check 3

Refer to trapezoid HIJK below, with HI KJ. Find x and y.

H I

93° y

x 79°

K J

Now Try Problem 29

	A	Upper base	B
		Upper base
	Leg	angles	Leg
			Leg

		Lower base
		angles
D		Lower base	C
D

		Trapezoid
		·		·
In the ﬁgure above, we can view AD as a transversal cutting the parallel lines AB
·	D are interior angles on the same side of a transversal, they are
and DC. Since A and
	·		·	·

supplementary. Similarly, BC is a transversal cutting the parallel lines AB and DC. Since

B and C are interior angles on the same side of a transversal, they are also supplementary. These observations lead us to the conclusion that there are always two pairs of supplementary angles in any trapezoid.

EXAMPLE 3
EXAMPLE 3	Refer to trapezoid KLMN below, with KL NM. Find x and y.
	K	L
	K	121°
	x	121°
	82°	y
	N		M

Strategy We will use the interior angles property twice to write two equations that mathematically model the situation.

WHY We can then solve the equations to ﬁnd x and y.

Solution K and N are interior angles on the same side of transversal KN that

cuts the parallel lines segments KL and NM. Similarly, L and M are interior

angles on the same side of transversal LM that cuts the parallel lines segments KL and NM. Recall that if two parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary. We can use this fact twice— once to ﬁnd x and a second time to ﬁnd y.

m( K) m( N) 180°	The sum of the measures of supplementary		17

	angles is 180°.		7 10
			180
x 82° 180°	Substitute x for m( K) and 82° for m( N).		82
			98
x 98°	To isolate x, subtract 82° from both sides.

Thus, x is 98°.
m( L) m( M) 180°	The sum of the measures of supplementary	7 10

		180
	angles is 180°.	121
121° y 180°	Substitute 121° for m( L) and y for m( M).	59

y 59°	To isolate y, subtract 121° from both sides.

Thus, y is 59°.
If the nonparallel sides of a trapezoid are the				D	E
same length, it is called an isosceles trapezoid.
The ﬁgure on the right shows isosceles trapezoid
DEFG with			. In an isosceles trapezoid,
	DG	EF
both pairs of base angles are congruent. In the				G		F
ﬁgure, D E and G F.					Isosceles trapezoid

EXAMPLE 4

9.6 Quadrilaterals and Other Polygons

771

Landscaping A cross section of a drainage ditch shown below is an isosceles trapezoid with AB DC. Find x and y.

A B

8 ft

120° y

D C

Self Check 4

Refer to the isosceles trapezoid shown below with RS UT. Find x and y.

R	S
x	10 in.
58°	y
U	T

Now Try Problem 31

Strategy We will compare the nonparallel sides and compare a pair of base angles of the trapezoid to ﬁnd each unknown.

WHY The nonparallel sides of an isosceles trapezoid have the same length and both pairs of base angles are congruent.

Solution Since AD and BC are the nonparallel sides of an isosceles trapezoid, m(AD) and m(BC) are equal, and x is 8 ft.

Since D and C are a pair of base angles of an isosceles trapezoid, they are congruent and m( D) m( C). Thus, y is 120°.

4Use the formula for the sum of the angle measures of a polygon.

In the ﬁgure shown below, a protractor was used to ﬁnd the measure of each angle of the quadrilateral. When we add the four angle measures, the result is 360°.

79°
	127°	2 3
	127°	2 3
		88
		79
		127
		66
88°	66°	360

88° + 79° + 127° + 66° = 360°

This illustrates an important fact about quadrilaterals: The sum of the measures of the angles of any quadrilateral is 360°.This can be shown using the diagram in ﬁgure

(a) on the following page. In the ﬁgure, the quadrilateral is divided into two triangles. Since the sum of the angle measures of any triangle is 180°, the sum of the measures of the angles of the quadrilateral is 2 180°, or 360°.

A similar approach can be used to ﬁnd the sum of the measures of the angles of any pentagon or any hexagon. The pentagon in ﬁgure (b) is divided into three triangles. The sum of the measures of the angles of the pentagon is 3 180°, or 540°. The hexagon in ﬁgure (c) is divided into four triangles.The sum of the measures of the angles of the hexagon is 4 180°, or 720°. In general, a polygon with n sides can be divided into n 2 triangles.Therefore, the sum of the angle measures of a polygon can be found by multiplying 180° by n 2.

Do the division.

To isolate n, divide both sides by 180°.

Do the additions.

Distribute the multiplication by 180°.

Substitute 1,080° for S, the sum of the measures of the angles.

This is the formula for the sum of the measures of the angles of a polygon.

EXAMPLE 6

772	Chapter 9 An Introduction to Geometry

Self Check 5

Find the sum of the angle measures of the polygon shown below.

Now Try Problem 33

Self Check 6

The sum of the measures of the angles of a polygon is 1,620°. Find the number of sides the polygon has.

Now Try Problem 41

Quadrilateral	Pentagon	Hexagon
	1	1
	1	2
	2	2
1	2
2	3	3
	3	4


2 • 180° = 360°	3 • 180° = 540°	4 • 180° = 720°
(a)	(b)	(c)

Sum of the Angles of a Polygon

The sum S, in degrees, of the measures of the angles of a polygon with n sides is given by the formula

S (n 2)180°

EXAMPLE 5 Find the sum of the angle measures of a 13-sided polygon.

Strategy We will substitute 13 for n in the formula S (n 2)180° and evaluate the right side.

WHY The variable S represents the unknown sum of the measures of the angles of the polygon.

Solution

S (n 2)180°	This is the formula for the sum of the measures	180

	of the angles of a polygon.	11
S (13 2)180°	Substitute 13 for n, the number of sides.	180
		1800
(11)180°	Do the subtraction within the parentheses.	1,980

1,980°	Do the multiplication.

The sum of the measures of the angles of a 13-sided polygon is 1,980°.

The sum of the measures of the angles of a polygon is 1,080°. Find the number of sides the polygon has.

Strategy We will substitute 1,080° for S in the formula S (n 2)180° and solve for n.

WHY The variable n represents the unknown number of sides of the polygon.

Solution

	S	(n 2)180°
1,080°		(n 2)180°
1,080°		180°n 360°
1,080° 360°		180°n 360° 360°
1,440°		180°n
1,440°			180°n

180°			180°
8		n

The polygon has 8 sides. It is an octagon.

To isolate 180°n, add 360° to both sides.

	1			8
1,080				8
1,080		180	1,440
	360	180	1,440
	360	1 440
1,440		1 440
1,440			0
			0

	9.6 Quadrilaterals and Other Polygons	773

ANSWERS TO SELF CHECKS
ANSWERS TO SELF CHECKS
1.	a. 90° b. 12 ft c. 6.5 ft d. 6.5 ft 3. 87°, 101° 4. 10 in., 58° 5. 900°
6.	11 sides

S E C T I O N 9.6

STUDY SET

VOCABULARY

Fill in the blanks.

is a polygon with four sides.

is a quadrilateral with opposite sides

parallel.

is a quadrilateral with four right angles.

A rectangle with all sides of equal length is a

is a parallelogram with four sides of

equal length.

A segment that joins two nonconsecutive vertices of a

polygon is called a

of the polygon.


7.	A			has two sides that are parallel and two
	sides that are not parallel. The parallel sides are called
		. The legs of an			trapezoid have the
	same length.
8.	A		polygon has sides that are all the same
	length and angles that are all the same measure.

CONCEPTS

9.Refer to the polygon below.

a. How many vertices does it have? List them.

11. A parallelogram is shown below. Fill in the blanks.

a.	ST			b.	SV		TU
			S			T
		V			U

12.Refer to the rectangle below.

a.How many right angles does the rectangle have? List them.

b.Which sides are parallel?

c.Which sides are of equal length?

d.Copy the ﬁgure and draw the diagonals. Call the point where the diagonals intersect point X. How many diagonals does the ﬁgure have? List them.

b.How many sides does it have? List them.

c.How many diagonals does it have? List them.

d.Tell which of the following are acceptable ways of naming the polygon.

quadrilateral ABCD		D
quadrilateral CDBA	A
quadrilateral ACBD	B	C
	B
quadrilateral BADC

10. Draw an example of each type of quadrilateral.

a.	rhombus	b.	parallelogram
c.	trapezoid	d.	square
e.	rectangle	f.	isosceles trapezoid

13. Fill in the blanks. In any rectangle:


a.	All four angles are	angles.
b.	Opposite sides are		.

c.	Opposite sides have equal			.
d.	The diagonals have equal			.

e.	The diagonals intersect at their				.

14. Refer to the ﬁgure below.

15.In the ﬁgure below, TR DF, DT FR, and

m( D) 90°. What type of quadrilateral is DTRF?

D T

F R

774Chapter 9 An Introduction to Geometry

16.Refer to the parallelogram shown below. If m(GI) 4 and m(HJ) 4, what type of ﬁgure is quadrilateral GHIJ?

J G

17.a. Is every rectangle a square?

b.Is every square a rectangle?

c.Is every parallelogram a rectangle?

d.Is every rectangle a parallelogram?

e.Is every rhombus a square?

f.Is every square a rhombus?

18.Trapezoid WXYZ is shown below. Which sides are parallel?

X Y

19.Trapezoid JKLM is shown below.

a.What type of trapezoid is this?

b.Which angles are the lower base angles?

c.Which angles are the upper base angles?

d.Fill in the blanks:

m( J) m( ) m( K) m( ) m(JK) m()

K L

20.Find the sum of the measures of the angles of the hexagon below.

110° 170°

105°

80°

160° 95°

NOTATION

21. What do the tick marks in the ﬁgure indicate?

A B

22.Rectangle ABCD is shown below. What do the tick marks indicate about point X?

A	B
	X
D	C

23.In the formula S (n 2)180°, what does S represent? What does n represent?

24.Suppose n 12. What is (n 2)180°?

GUIDED PRACTICE

In Problems 25 and 26, classify each quadrilateral as a rectangle, a square, a rhombus, or a trapezoid. Some ﬁgures may be correctly classiﬁed in more than one way. See Objective 1.

25. a.		4 in.	b.
	4 in.		4 in.
	4 in.		4 in.

4 in.

c. d.

26. a.

8 cm

8 cm	8 cm
8 cm
	8 cm

c. d.

27. Rectangle ABCD is shown below. See Example 1.


a.	What is m( DCB)?						D		C


	What is m(					)?
b.			AX					9
	What is m(			)?
c.		AC						X
	What is m(				)?
d.			BD				A		B

28. Refer to rectangle EFGH shown below. See Example 1.

a.	Find m( EHG).				b.	Find m(FH).
	Find m(			).		Find m(			).
c.	Find m(		EI	).	d.	Find m(	EG		).
	F							G
		16			I
	E				I			H

D C

29.	Refer to the trapezoid shown below. See Example 3.
	a.	Find x.		b.	Find y.
		138°			y
			x		85°
30.	Refer to trapezoid MNOP shown below. See Example 3.
	a.	Find m( O).		b.	Find m( M).
		M		119.5°	N
				119.5°

		P			O
		P			O

31.Refer to the isosceles trapezoid shown below.

See Example 4.

a.	Find m(BC).	b.	Find x.
c.	Find y.	d.	Find z.
	A	22	B
	A		B
	y		z
	9
	x		70°
	D		C

32.Refer to the trapezoid shown below. See Example 4.

a.Find m( T).

b.Find m( R).

c.Find m( S).

Q T

47.5°

R S

Find the sum of the angle measures of the polygon.

See Example 5.

33.a 14-sided polygon

34.a 15-sided polygon

35.a 20-sided polygon

36.a 22-sided polygon

37.an octagon

38.a decagon

39.a dodecagon

40.a nonagon

9.6 Quadrilaterals and Other Polygons

775

Find the number of sides a polygon has if the sum of its angle measures is the given number. See Example 6.

41.	540°	42.	720°
43.	900°	44.	1,620°
45.	1,980°	46.	1,800°
47.	2,160°	48.	3,600°

TRY IT YOURSELF

49.Refer to rectangle ABCD shown below.

a.Find m( 1).

b.Find m( 3).

c.Find m( 2).

d.If m(AC) is 8 cm, ﬁnd m(BD).

e.Find m(PD).

D		C
D	2	C
	2
	3

1 P

60°

50.The following problem appeared on a quiz. Explain why the instructor must have made an error when typing the problem.

The sum of the measures of the angles of a polygon is 1,000°. How many sides does the polygon have?

For Problems 51 and 52, ﬁnd x. Then ﬁnd the measure of each angle of the polygon.

51.A

	2x 10°	B
		3x 30°
	2x	x
	D	C
52.	A	B

x 12°

G x

x 8° C

	x
F	x 50°	x

E D

776	Chapter 9 An Introduction to Geometry

APPLICATIONS

53.QUADRILATERALS IN EVERYDAY LIFE What quadrilateral shape do you see in each of the following objects?

a. podium (upper portion) b. checkerboard

c. dollar bill

d. swimming ﬁn

	e. camper shell window
	54. FLOWCHART A ﬂowchart
	shows a sequence of steps to be	Start
	performed by a computer to
	solve a given problem. When	Open the
	designing a ﬂowchart, the	Files
	designing a ﬂowchart, the
	programmer uses a set of	Read
	standardized symbols to	Read
	standardized symbols to	a Record
	represent various operations to
	be performed by the computer.	More
	Locate a rectangle, a rhombus,	More
	Locate a rectangle, a rhombus,	Records to Be
	and a parallelogram in the	Records to Be
	and a parallelogram in the	Processed?
	ﬂowchart shown to the right.
		Close the
		Files
		End

55.BASEBALL Refer to the ﬁgure to the right. Find the sum of the measures of the angles of home plate.

56.TOOLS The utility knife blade shown below has the

shape of an isosceles trapezoid. Find x, y, and z.

	1
	1
			1– in.
			1– in.
	4
					z
x				3
x						– in.
65°				4
65°						y
						y
				3
				3
		2		– in.
		2		– in.
				8

WRITING

57.Explain why a square is a rectangle.

58.Explain why a trapezoid is not a parallelogram.

59.MAKING A FRAME After gluing and nailing the pieces of a picture frame together, it didn’t look right to a frame maker. (See the ﬁgure to the right.) How can she use a tape measure

to make sure the corners are 90° (right) angles?

60.A decagon is a polygon with ten sides. What could you call a polygon with one hundred sides? With one thousand sides? With one million sides?

REVIEW

Write each number in words.

61.254,309

62.504,052,040

63.82,000,415

64.51,000,201,078

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