Proving Triangles Similar LESSON 7-3 Additional Examples

MX AB

. Explain why the triangles are similar. Write a similarity statement. Because

MX AB

, 

AXM



AXM



BXK

.

and 

BXK

are both right angles, so 

A



B

because their measures are equal.

AMX

~

BKX

by the Angle-Angle Similarity Postulate (AA ~ Postulate).

Quick Check HELP GEOMETRY

Proving Triangles Similar LESSON 7-3 Additional Examples

Explain why the triangles must be similar. Write a similarity statement.



YVZ



WVX

because they are vertical angles.

VY VW

= 12 = 24 1 2 and

VZ VX

= 18 36 1 = , so corresponding sides are proportional.

2 Therefore,

YVZ

~

WVX

by the Side-Angle-Side Similarity Theorem (SAS Similarity Theorem).

Quick Check HELP GEOMETRY

Proving Triangles Similar LESSON 7-3 Additional Examples

ABCD

is a parallelogram. Find

WY

. Because

ABCD

is a parallelogram,

AB

||

DC

.



XAW



ZYW

and 

AXW



YZW

because parallel lines cut by a transversal form congruent alternate interior angles.

Therefore,

AWX ~ YWZ

by the AA ~ Postulate. Use the properties of similar triangles to find

WY

.

WY WA

=

WZ WX WY

5

WY

10 = 4 10 =  5

WY

= 12.5

Corresponding sides of ~ triangles are proportional.

Substitute.

Solve for

WY

.

Quick Check HELP GEOMETRY

Proving Triangles Similar LESSON 7-3 Additional Examples

Joan places a mirror 24 ft from the base of a tree. When she stands 3 ft from the mirror, she can see the top of the tree reflected in it. If her eyes are 5 ft above the ground, how tall is the tree?

Draw the situation described by the example.

TR

represents the height of the tree, point

M

mirror, and point

J

represents Joan’s eyes. represents the Both Joan and the tree are perpendicular to the ground, so

m



JOM

=

m



TRM

, and therefore 

JOM



TRM.

The light reflects off a mirror at the same angle at which it hits the mirror, so 

JMO



TMR

.

Use similar triangles to find the height of the tree.

HELP GEOMETRY

Proving Triangles Similar LESSON 7-3 Additional Examples (continued)

JOM

~

TRM

TR JO TR

5

TR

5 =

RM OM

= 24 3 24 =  5

TR

= 40 The tree is 40 ft tall. AA ~ Postulate Corresponding sides of ~ triangles are proportional. Substitute. Solve for

TR

.

Quick Check HELP GEOMETRY