10.2 Proving Triangles Similar

Geometry

Mr. Calise

Objectives/Assignment

Identify similar triangles.Use similar triangles in real-life problems

such as using shadows to determine the height of the Great Pyramid

Identifying Similar Triangles

In this lesson, you will continue the study similar polygons by looking at the properties of similar triangles.

Ex. 1: Writing Proportionality Statements

In the diagram, ∆BTW ~ ∆ETC.

a. Write the statement of proportionality.

b. Find mTEC.c. Find ET and BE.

12

203

T

B W

E C

79°

34°

Ex. 1: Writing Proportionality Statements

In the diagram, ∆BTW ~ ∆ETC.

a. Write the statement of proportionality.

12

203

T

B W

E C

79°

34°

ET

BT

TC

TW

CE

WB= =

Ex. 1: Writing Proportionality Statements

In the diagram, ∆BTW ~ ∆ETC.

b. Find mTEC.B TEC, SO

mTEC = 79°

12

203

T

B W

E C

79°

34°

Ex. 1: Writing Proportionality Statements

In the diagram, ∆BTW ~ ∆ETC.

c. Find ET and BE.

12

203

T

B W

E C

79°

34°

CEWB

ETBT

=

312

ET20

=

3(20)12

= ET

ET=5

Write proportion.

Substitute values.

Multiply each side by 20.

Simplify.

Because BE = BT – ET, BE = 20 – 5 = 15. So, ET is 5 units and BE is 15 units.

Postulate 25 Angle-Angle Similarity Postulate

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

If JKL XYZ and KJL YXZ, then ∆JKL ~ ∆XYZ.

K

J

L

Y

X

Z

Ex. 2: Proving that two triangles are similar

Color variations in the tourmaline crystal shown lie along the sides of isosceles triangles. In the triangles, each vertex measures 52°. Explain why the triangles are similar.

Ex. 2: Proving that two triangles are similar

Solution. Because the triangles are isosceles, you can determine that each base angle is 64°. Using the AA Similarity Postulate, you can conclude the triangles are similar.

Side-Angle-Side Similarity TheoremSide-Angle-Side Similarity Theorem

If an angle in one triangle is congruent to an angle in another triangle, and the sides including the two angles are proportional, then the two triangles are similar.

(SAS Similarity Thm.)

USING SIMILARITY THEOREMS

THEOREM S

THEOREM 10.1 Side-Angle-Side (SAS) Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

then XYZ ~ MNP.

ZXPM

XYMN

If X M and =

X

Z Y

M

P N

Side-Side-Side Similarity TheoremSide-Side-Side Similarity Theorem

If the corresponding sides of two triangles are proportional, then the two triangles are similar.

(SSS Similarity Thm.)

USING SIMILARITY THEOREMS

THEOREM S

THEOREM 10.2 Side-Side-Side (SSS) Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

If = =A BPQ

BCQR

CARP

then ABC ~ PQR.

A

B C

P

Q R

E

F D8

6 4A C

B

12

6 9

G J

H

14

6 10

Using the SSS Similarity Theorem

Which of the following three triangles are similar?

SOLUTION To decide which of the triangles are similar, consider the

ratios of the lengths of corresponding sides.

Ratios of Side Lengths of ABC and DEF

= = , 6 4

AB DE

3 2

Shortest sides

= = , 12 8

CA FD

3 2

Longest sides

= = 9 6

BC EF

3 2

Remaining sides

Because all of the ratios are equal, ABC ~ DEF

Finding Distance Indirectly

Similar triangles can be used to find distances that are difficult to measure directly.

ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.

85 ft6.5 ft

5 ft

A

B

C E

DUse similar triangles to estimate the height of the wall.

Not drawn to scale

Finding Distance Indirectly

85 ft6.5 ft

5 ft

A

B

C E

D

Use similar triangles to estimate the height of the wall.

SOLUTION

Using the fact that ABC and EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.

Due to the reflective property of mirrors, you can reason that ACB ECD.

85 ft6.5 ft

5 ft

A

B

C E

D

DE65.38

Finding Distance Indirectly

Use similar triangles to estimate the height of the wall.

SOLUTION

= ECAC

DEBA

Ratios of lengths of corresponding sides are equal.

Substitute.

Multiply each side by 5 and simplify.

DE5

= 856.5

So, the height of the wall is about 65 feet.

Note:

If two polygons are similar, then the ratio of any two corresponding lengths (such as altitudes, medians, angle bisector segments, and diagonals) is equal to the scale factor of the similar polygons.

Ex. 5: Using Scale Factors

Find the length of the altitude QS. Solution: Find the scale factor of

∆NQP to ∆TQR.

NP

TR

12+12

8 + 8= 24

1632

==

Now, because the ratio of the lengths of the altitudes is equal to the scale factor, you can write the following equation:

QM

QS=

3

2

Substitute 6 for QM and solve for QS to show that QS = 4

1212

6

88 S

MN P

Q

R T

Homework

Finish The Worksheets from ThursdayDUE on Monday!