Transcript 7-3 Proving Triangles Similar

Proving Triangles Similar
LESSON 7-3
Additional Examples
MX AB. Explain why the triangles are similar.
Write a similarity statement.
Because MX AB, AXM and BXK are both right angles, so
AXM BXK.
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 12 1
VZ 18 1
=
=
and
= = , so corresponding sides are proportional.
VW 24 2
VX 36 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
WZ
=
WA
WX
10
WY
= 4
5
10
WY = 4  5
WY = 12.5
HELP
Corresponding sides of ~ triangles are proportional.
Substitute.
Solve for WY.
Quick Check
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 represents the
mirror, and point J represents Joan’s eyes.
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
RM
TR
=
OM
JO
TR
= 24
3
5
24
TR
=
3 5
5
TR = 40
The tree is 40 ft tall.
HELP
AA ~ Postulate
Corresponding sides of ~ triangles are proportional.
Substitute.
Solve for TR.
Quick Check
GEOMETRY