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754	Chapter 9 An Introduction to Geometry

Objectives

1Identify corresponding parts of congruent triangles.

2Use congruence properties to prove that two triangles are congruent.

3Determine whether two triangles are similar.

4Use similar triangles to ﬁnd unknown lengths in application problems.

S E C T I O N 9.5

Congruent Triangles and Similar Triangles

In our everyday lives, we see many types of triangles.Triangular-shaped kites, sails, roofs, tortilla chips, and ramps are just a few examples. In this section, we will discuss how to compare the size and shape of two given triangles. From this comparison, we can make observations about their side lengths and angle measures.

1 Identify corresponding parts of congruent triangles.

Simply put, two geometric ﬁgures are congruent if they have the same shape and size. For example, if ABC and DEF shown below are congruent, we can write

ABC DEF Read as “Triangle ABC is congruent to triangle DEF.”

C F

A B D E

One way to determine whether two triangles are congruent is to see if one triangle can be moved onto the other triangle in such a way that it ﬁts exactly. When we writeABC DEF, we are showing how the vertices of one triangle are matched to the vertices of the other triangle to obtain a “perfect ﬁt.”We call this matching of points a correspondence.




ABC DEF

A 4 D Read as “Point A corresponds to point D.”

B 4 E Read as “Point B corresponds to point E.”

C 4 F Read as “Point C corresponds to point F.”

When we establish a correspondence between the vertices of two congruent triangles, we also establish a correspondence between the angles and the sides of the triangles. Corresponding angles and corresponding sides of congruent triangles are called corresponding parts. Corresponding parts of congruent triangles are always congruent. That is, corresponding parts of congruent triangles always have the same measure. For the congruent triangles shown above, we have

m( A) m( D)					m( B) m( E)
m(		) m(		)	m(		) m(		)
	BC		EF			AC		DF

m( C) m( F)

m(AB) m(DE)

Congruent Triangles

Two triangles are congruent if and only if their vertices can be matched so that the corresponding sides and the corresponding angles are congruent.

	EXAMPLE 1	Refer to the ﬁgure below, where XYZ PQR.
a. Name the six congruent corresponding parts
	of the triangles.		Z	R
b. Find m( P).		27°	5 in.	88°	11 in.

c. Find m(XZ).			Y		P
c. Find m(XZ).		X	Y	Q	P

9.5 Congruent Triangles and Similar Triangles

755

Strategy We will establish the correspondence between the vertices of XYZ and the vertices of PQR.

WHY This will, in turn, establish a correspondence between the congruent corresponding angles and sides of the triangles.

Solution

a. The correspondence between the vertices is

XYZ PQR

X 4 P Y 4 Q

Z 4 R

Corresponding parts of congruent triangles are congruent. Therefore, the congruent corresponding angles are

X P Y Q Z R

The congruent corresponding sides are

YZ QR XZ PR XY PQ

b.From the ﬁgure, we see that m( X) 27°. Since X P, it follows that m( P) 27°.

c.From the ﬁgure, we see that m(PR) 11 inches. Since XZ PR, it follows

that m(XZ) 11 inches.

2Use congruence properties to prove that two triangles are congruent.

Sometimes it is possible to conclude that two triangles are congruent without having to show that three pairs of corresponding angles are congruent and three pairs of corresponding sides are congruent.To do so, we apply one of the following properties.

SSS Property

If three sides of one triangle are congruent to three sides of a second triangle, the triangles are congruent.

We can show that the triangles shown below are congruent by the SSS property:

	D			S
	D
3	4		5	3
3	4			3
C	E	R		T
	5			4

CD ST Since m(CD) 3 and m(ST) 3, the segments are congruent.

DE TR Since m(DE) 4 and m(TR) 4, the segments are congruent.

EC RS Since m(EC) 5 and m(RS) 5, the segments are congruent.

Therefore, CDE STR.

Self Check 1

Refer to the ﬁgure below, where

ABC EDF.

a.Name the six congruent corresponding parts of the triangles.

b.Find m( C).

c.Find m(FE).

C		F
3 ft	7 ft	20°
3 ft	110°
	110°
A	B D	E

Now Try Problem 33

SAS Property

If two sides and the angle between them in one triangle are congruent, respectively, to two sides and the angle between them in a second triangle, the triangles are congruent.

756	Chapter 9 An Introduction to Geometry

We can show that the triangles shown below are congruent by the SAS property:

		E
V
2 90°	3	3
2 90°	3		90°
T	U	G	90°	F
T	U	G	2	F
			2

TV FG Since m(TV) 2 and m(FG) 2, the segments are congruent.

V G Since m( V) 90° and m( G) 90°, the angles are congruent.

UV EG Since m(UV) 3 and m(EG) 3, the segments are congruent.

Therefore, TVU FGE.

ASA Property

If two angles and the side between them in one triangle are congruent, respectively, to two angles and the side between them in a second triangle, the triangles are congruent.

We can show that the triangles shown below are congruent by the ASA property:

R		C
9 82°		82°	9
60°	Q	60°	B
P	Q	A	B

P B Since m( P) 60° and m( B) 60°, the angles are congruent.

PR BC Since m(PR) 9 and m(BC) 9, the segments are congruent.

R C Since m( R) 82° and m( C) 82°, the angles are congruent.

Therefore, PQR BAC.

Caution! There is no SSA property. To illustrate this, consider the triangles shown below. Two sides and an angle of ABC are congruent to two sides and an angle of DEF. But the congruent angle is not between the congruent sides.

We refer to this situation as SSA. Obviously, the triangles are not congruent because they are not the same shape and size.

		B	E	The tick marks indicate congruent
				parts. That is, the sides with one tick
				mark are the same length, the sides
				with two tick marks are the same
				length, and the angles with one tick
A	C	D	F	mark have the same measure.

EXAMPLE 2 Explain why the triangles in the ﬁgure on the following page are congruent.

Strategy We will show that two sides and the angle between them in one triangle are congruent, respectively, to two sides and the angle between them in a second triangle.

9.5 Congruent Triangles and Similar Triangles

757

WHY Then we know that the two triangles are congruent by the SAS property.

Solution Since vertical angles are congruent,

1 2

From the ﬁgure, we see that

AC EC and BC DC

Since two sides and the angle between them in one triangle are congruent, respectively, to two sides and the angle between them in a second triangle,ABC EDC by the SAS property.

B
	10 cm
A	1	C 5 cm
A	5 cm	E
	5 cm	2
		10 cm

EXAMPLE 3 Are RST and RUT in the ﬁgure on the right congruent?

Strategy We will show that two angles and the side between them in one triangle are congruent, respectively, to two angles and the

side between them in a second triangle.

WHY Then we know that the two triangles are congruent by the ASA property.

Solution From the markings on the ﬁgure, we know that two pairs of angles are congruent.

SRT URT These angles are marked with 1 tick mark, which indicates that they have the same measure.

STR UTR These angles are marked with 2 tick marks, which indicates that they have the same measure.

From the ﬁgure, we see that the triangles have side RT in common. Furthermore, RT is between each pair of congruent angles listed above. Since every segment is congruent to itself, we also have

RT RT

Knowing that two angles and the side between them in RST are congruent, respectively, to two angles and the side between them in RUT, we can conclude that RST RUT by the ASA property.

3 Determine whether two triangles are similar.

We have seen that congruent triangles have the same shape and size. Similar triangles have the same shape, but not necessarily the same size. That is, one triangle is an exact scale model of the other triangle. If the triangles in the ﬁgure below are similar, we can write ABC DEF (read the symbol as “is similar to”).

Self Check 2

Are the triangles in the ﬁgure below congruent? Explain why or why not.

Now Try Problem 35

Self Check 3

Are the triangles in the following ﬁgure congruent? Explain why or why not.

Q S

Now Try Problem 37

A B D E

Success Tip Note that congruent triangles are always similar, but similar triangles are not always congruent.

EXAMPLE 4

758	Chapter 9 An Introduction to Geometry

Self Check 4

If GEF IJH, name the congruent angles and the sides that are proportional.

G E J I

Now Try Problem 39

The formal deﬁnition of similar triangles requires that we establish a correspondence between the vertices of the triangles. The deﬁnition also involves the word proportional.

Recall that a proportion is a mathematical statement that two ratios (fractions) are equal. An example of a proportion is

12 48

In this case, we say that 12 and 48 are proportional.

Similar Triangles

Two triangles are similar if and only if their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are proportional.

Refer to the ﬁgure below. If PQR CDE, name the congruent angles and the sides that are proportional.

Q R D E

Strategy We will establish the correspondence between the vertices of PQR and the vertices of CDE.

WHY This will, in turn, establish a correspondence between the congruent corresponding angles and proportional sides of the triangles.

Solution When we write PQR CDE,a correspondence between the vertices of the triangles is established.




	PQR CDE

Since the triangles are similar, corresponding angles are congruent:

P C Q D R E

The lengths of the corresponding sides are proportional. To simplify the notation, we will now let PQ m(PQ), CD m(CD), QR m(QR), and so on.

PQ		QR		QR		PR		PQ		PR
CD		DE		DE		CE		CD		CE

Written in a more compact way, we have

CDPQ QRDE CEPR

Property of Similar Triangles

If two triangles are similar, all pairs of corresponding sides are in proportion.

9.5 Congruent Triangles and Similar Triangles

759

It is possible to conclude that two triangles are similar without having to show that all three pairs of corresponding angles are congruent and that the lengths of all three pairs of corresponding sides are proportional.

AAA Similarity Theorem

If the angles of one triangle are congruent to corresponding angles of another triangle, the triangles are similar.

	EXAMPLE 5
	EXAMPLE 5	In the ﬁgure on the right, PR MN. Are PQR and NQM
similar triangles?				M
				M
Strategy We will show that the angles of P
one triangle are congruent to corresponding			Q	N
angles of another triangle.			Q	N
angles of another triangle.

WHY Then we know that the two triangles

are similar by the AAA property.

Solution Since vertical angles are congruent,

PQR NQM This is one pair of congruent corresponding angles.

In the ﬁgure, we can view PN as a transversal cutting parallel line segments PR and MN. Since alternate interior angles are then congruent, we have:

RPQ MNQ This is a second pair of congruent corresponding angles.

Furthermore, we can view RM as a transversal cutting parallel line segments PR and MN. Since alternate interior angles are then congruent, we have:

QRP QMN This is a third pair of congruent corresponding angles.

These observations are summarized in		M
the ﬁgure on the right. We see that	P
corresponding angles of PQR are	P
corresponding angles of PQR are
congruent to corresponding angles of	Q	N
NQM. By the AAA similarity theorem,
we can conclude that
PQR NQM	R

	EXAMPLE 6	In the ﬁgure below, RST JKL. Find:
Strategy To ﬁnd x, we will write a			T
proportion of corresponding sides so that

x is the only unknown.Then we will solve
the proportion for x.We will use a similar			x
method to ﬁnd y.			48
WHY Since RST JKL, we know			S
that the lengths of corresponding sides of			36
RST and JKL are proportional.			R

a. x

b. y

32 20

Solution

a.When we write RST JKL, a correspondence between the vertices of the two triangles is established.

RST JKL

Self Check 5

In the ﬁgure below, YA ZB. Are XYA and XZB similar triangles?

	Z
	Y
X	B
	A

Now Try Problems 41 and 43

Self Check 6

In the ﬁgure below,

DEF GHI. Find:

a. x

b. y

15	18	y	13.5

D	G
D	4.5	H
	4.5
	x
	x
	E

Now Try Problem 53

EXAMPLE 7

760	Chapter 9 An Introduction to Geometry

The lengths of corresponding sides of these similar triangles are proportional.

RTJL KLST

4832 20x

48(20) 32x

960 32x

30 x

Thus, x is 30.

Each fraction is a ratio of a side length of RST to its corresponding side length of JKL.

Substitute: RT 48, JL 32, ST x, and KL 20.

Find each cross product and set them equal.

Do the multiplication.

To isolate x, undo the multiplication by 32 by dividing both sides by 32.

4820 960

32 960

b.To ﬁnd y, we write a proportion of corresponding side lengths in such a way that y is the only unknown.

RTJL RSJK

4832 36y

48y 32(36)

48y 1,152 y 24

Thus, y is 24.

Substitute: RT 48, JL 32, RS 36, and JK y.

Find each cross product and set them equal.

Do the multiplication.

To isolate y, undo the multiplication by 48 by dividing both sides by 48.

48 1,152

1080

192

1152

192 0

Self Check 7

In the ﬁgure below,

ABC EDC. Find h.

25 ft

40 ft

C 2 ft D

Now Try Problem 85

4Use similar triangles to ﬁnd unknown lengths in application problems.

Similar triangles and proportions can be used to ﬁnd lengths that would normally be difﬁcult to measure. For example, we can use the reﬂective properties of a mirror to calculate the height of a ﬂagpole while standing safely on the ground.

To determine the height of a ﬂagpole, a woman walks to a point 20 feet from its base, as shown below.Then she takes a mirror from her purse, places it on the ground, and walks 2 feet farther away, where she can see the top of the pole reﬂected in the mirror. Find the height of the pole.

The woman’s eye level is 5 feet from the ground.

5 ft

E A 2 ft 20 ft

Strategy We will show that ABC EDC.

WHY Then we can write a proportion of corresponding sides so that h is the only unknown and we can solve the proportion for h.

Solution To show that ABC EDC, we begin by applying an important fact about mirrors.When a beam of light strikes a mirror, it is reﬂected at the same angle as it hits the mirror. Therefore, BCA DCE. Furthermore, A E because the woman and the ﬂagpole are perpendicular to the ground. Finally, if two pairs of

9.5 Congruent Triangles and Similar Triangles

761

corresponding angles are congruent, it follows that the third pair of corresponding angles are also congruent: B D. By the AAA similarity theorem, we conclude that ABC EDC.

Since the triangles are similar, the lengths of their corresponding sides are in proportion. If we let h represent the height of the ﬂagpole, we can ﬁnd h by solving the following proportion.

Height of the ﬂagpole

Height of the woman

2h h

5(20)

100

Distance from ﬂagpole to mirror

Distance from woman to mirror

Find each cross product and set them equal.

Do the multiplication.

To isolate h, divide both sides by 2.

The ﬂagpole is 50 feet tall.

ANSWERS TO SELF CHECKS

a. A E, B D, C F,

b. 20° c. 3 ft

yes, by the SAS property 3.

yes, by the SSS property

4. G I, E J,

F H; EGJI GFIH , GFIH FEHJ, EGJI

FEHJ 5. yes, by the AAA similarity theorem:

X X, XYA XZB, XAY XBZ 6. a. 6

b. 11.25 7. 500 ft

S E C T I O N 9.5 STUDY SET

VOCABULARY

Fill in the blanks.

1.triangles are the same size and the same

shape.

2.When we match the vertices of ABC with the vertices of DEF, as shown below, we call this

	matching of points a								.
	A 4 D	B 4 E					C 4 F
3.	Two angles or two line segments with the same
	measure are said to be							.
4.	Corresponding				of congruent triangles are
	congruent.
5.	If two triangles are						, they have the same shape
	but not necessarily the same size.
6.	A mathematical statement that two ratios (fractions)
	are equal, such as		x	4		, is called a				.
	are equal, such as			4		, is called a				.
		18		9

CONCEPTS

7. Refer to the triangles below.

a.Do these triangles appear to be congruent? Explain why or why not.

b.Do these triangles appear to be similar? Explain why or why not.

8.a. Draw a triangle that is

congruent to CDE shown below. Label it

ABC.

b.Draw a triangle that is similar to, but not congruent to,CDE. Label it MNO.

Fill in the blanks.

9. XYZ

Z P

762Chapter 9 An Introduction to Geometry

10.DEF

	A			D
			C F
	B			E
11.	RST
R		M
R
	T	S		N
	T		O	N
			O
12.		TAC
				T
	B
		5	10	6
	3			6


	D	E	C	A
	D	4	8	A

13.Name the six corresponding parts of the congruent triangles shown below.

Y T

Z A R B

14.Name the six corresponding parts of the congruent triangles shown below.

3 in.

5 in.

4 in.

5 in.

4 in.

3 in.

Fill in the blanks.

15.

Two triangles are

if and only if their

vertices can be matched so that the corresponding

sides and the corresponding angles are congruent.

16.

SSS property: If three

of one triangle are

congruent to three

of a second triangle, the

triangles are congruent.

17. SAS property: If two sides and the

between

them in one triangle are congruent, respectively, to

two sides and the

between them in a second

triangle, the triangles are congruent.

18.	ASA property: If two angles and the									between
	them in one triangle are congruent, respectively, to
	two angles and the			between them in a second
	triangle, the triangles are congruent.
Solve each proportion.
	x	20	5				35
19.			20.
19.	15	3	20.		8			x
	h	27	11.2						h
21.			22.
21.	2.6	13	22.			4			6

Fill in the blanks.

23.Two triangles are similar if and only if their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are

24.If the angles of one triangle are congruent to corresponding angles of another triangle, the triangles


	are	.
25.	Congruent triangles are always similar, but similar
	triangles are not always			.
26.	For certain application problems, similar triangles and
			can be used to ﬁnd lengths that would
	normally be difﬁcult to measure.

NOTATION

Fill in the blanks.
27.	The symbol is read as “	.”

28.	The symbol is read as “	.”

29.Use tick marks to show the congruent parts of the triangles shown below.

K H KR HJ M E

K M H E

R J

30.Use tick marks to show the congruent parts of the triangles shown below.

P T		LP	RT	FP		ST
	P				T
L		F R				S

GUIDED PRACTICE

Name the six corresponding parts of the congruent triangles.

See Objective 1.

C F

31.		AC

		DE
				A			D
			BC
	A
		E
			F		B		E

32.		AB				E

			EC

		AC			5 cm		D
	D					2	D
	D					2	4 cm
		B			4 cm	C	4 cm
		B			1	C
					1

1		A
1		5 cm
		5 cm
		B

33.Refer to the ﬁgure below, where BCD MNO.

a.Name the six congruent corresponding parts of the triangles. See Example 1.

b.Find m( N).

c.Find m(MO).

d.Find m(CD).

72°

9 ft

49°

B D O M

10 ft

34. Refer to the ﬁgure below, where DCG RST.

a. Name the six congruent corresponding parts of the triangles. See Example 1.

b. Find m( R). c. Find m(DG). d. Find m(ST).

C			S
54°	3 in.
	3 in.
60°			66°
D	G	T	2 in.	R

9.5 Congruent Triangles and Similar Triangles

763

Determine whether each pair of triangles is congruent. If they are, tell why. See Examples 2 and 3.

35.	36.
6 cm	cm	cm
3 cm	cm	cm
	6	6
5 cm

	5 cm		5 cm
5 cm	5 cm		5 cm
5 cm
3 cm
6 cm

37. 38.

6 m

39.Refer to the similar triangles shown below.

See Example 4.

a.Name 3 pairs of congruent angles.

b.Complete each proportion.

c.We can write the answer to part b in a more compact form:

LM MR HE

L R H E

40.Refer to the similar triangles shown below.

See Example 4.

a.Name 3 pairs of congruent angles.

b.Complete each proportion.

c.We can write the answer to part b in a more compact form:

DF YX WX

Y F W D

764	Chapter 9 An Introduction to Geometry

Tell whether the triangles are similar. See Example 5.

41.

42.

43. 44.

45.

70°

40°

70°

46.

47. 48.

49.	XY	ZD		50.	QR		TU
	Z		X			Q		R
	Z
		E						S
D					T			U
D			Y
			Y

51. 52.

In Problems 53 and 54, MSN TPR. Find x and y.

See Example 6.

53.S

28 10 x

M 21 N T 6 R

54.S

			P
12		y	6	x
			6

			R
	6	N	4	T
M	6
M
In Problems 55 and 56, MSN TPN. Find x and y.
See Example 6.
55.			P	56. M		40	S
M
	40	57					y
	40				32
50
			75
							N
	N
	N				18
	y	x			18		x
S		x
S					P		T
			T		P	24	T
						24

TRY IT YOURSELF

Tell whether each statement is true. If a statement is false, tell why.

57.If three sides of one triangle are the same length as the corresponding three sides of a second triangle, the triangles are congruent.

58.If two sides of one triangle are the same length as two sides of a second triangle, the triangles are congruent.

59.If two sides and an angle of one triangle are congruent, respectively, to two sides and an angle of a second triangle, the triangles are congruent.

60.If two angles and the side between them in one triangle are congruent, respectively, to two angles and the side between them in a second triangle, the triangles are congruent.

Determine whether each pair of triangles are congruent. If they are, tell why.

61.

62.

40°

63. 64.

40°	40°
6 yd	6 yd

65.

66.

31°

67. 68.

In Problems 69 and 70, ABC DEF. Find x and y.

69.				D
		C
		80°	y
			3 yd	4 yd
	A		B
				x
			E	F
			2 yd
70.					F
			C	x	20°
			C	25°	135°
		12 in.	D	25°	135°
		12 in.	D		E
			5 in.		E
		y	5 in.
	A	y
	A	8 in.	B
			B

In Problems 71 and 72, ﬁnd x and y.
71.	ABC ABD
			A
		y	55°	14 m
		y		14 m
		19°	B	x
	C	11 m			D
	C				D
72.	ABC DEC
		A			D
		37°	C	y
			C
		10 mi	46°
			x
		B	8 mi	E
		B		E
In Problems 73–76, ﬁnd x.
73.			74.	x cm
	5 mm	6 mm		x cm
	5 mm		cm	cm
					9 cm
	x mm				9 cm
	x mm	5 mm	7	7
		5 mm

9.5		Congruent Triangles and Similar Triangles		765
75.			76.	50°


7 in.			5 in.
7 in.		7 in.	5 in.	5 in.
				5 in.
50°			x
50°			x
	x
7 in.		7 in.

77.If DE in the ﬁgure below is parallel to AB, ABC will be similar to DEC. Find x.

78.If SU in the ﬁgure below is parallel to TV, SRU will be similar to TRV. Find x.

R
2
S		U
S	3	U
12	3
12
T		x	V
		x

79.If DE in the ﬁgure below is parallel to CB, EAD will be similar to BAC. Find x.

		D
		12	15
		12
A	x	E 4	B
	x	E 4

80.If HK in the ﬁgure below is parallel to AB, HCK will be similar to ACB. Find x.

H		K
H
		18
	6
	A	B
	A

5 cm

766	Chapter 9 An Introduction to Geometry

APPLICATIONS

81.SEWING The pattern that is sewn on the rear

pocket of a pair of blue jeans is shown below. IfAOB COD, how long is the stitching from

point A to point D?

A C

9.5 cm

8 cm

82.CAMPING The base of the tent pole is placed at the midpoint between the stake at point A and the stake

at point B, and it is perpendicular to the ground, as shown below. Explain why ACD BCD.

84.HEIGHT OF A BUILDING A man places a mirror on the ground and sees the reﬂection of the top of a building, as shown below. Find the height of the building.

	h
6 ft
8 ft	48 ft

85.HEIGHT OF A TREE The tree shown below casts a shadow 24 feet long when a man 6 feet tall casts a shadow 4 feet long. Find the height of the tree.

A D B

	83. A surveying crew needs to	from Campus to Careers
	ﬁnd the width of the river	Surveyor
	shown in the illustration
	below. Because of a	Laska
	dangerous current, they	Laska
	dangerous current, they	iStockphoto.com/Lukaz
	decide to stay on the west
	side of the river and use
	geometry to ﬁnd its width.
	Their approach is to create
	Their approach is to create	©
		©

two similar right triangles on

dry land. Then they write and solve a proportion to ﬁnd w. What is the width of the river?

20 ft	25 ft

74 ft

West East w ft

6 ft

4 ft

24 ft

86.WASHINGTON, D.C. The Washington Monument

casts a shadow of 16612 feet at the same time as a 5-foot-tall tourist casts a shadow of 112 feet. Find the height of the monument.

		h
5 ft
1	1	166	1
1	– ft	166	– ft
	2		2

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