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7.3 Proving Triangles Similar using AA, SSS, SAS.

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7.3 Proving Triangles Similarusing AA, SSS, SAS.

Angle-Angle (AA) ~ PostulateIf two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

W

Wis reflexive and both triangles have a right .

ΔWVX ~ΔWZY by AA ~ Post.

X

V

Y

Z

W

In the diagram, ΔWZY ~ ΔWVX by AA ~ Postulate

Find x.

Set up a proportion.

Cross multiply.

Distribute and foil.

16x + 8 = x2 + 10x + 16

0 = x2 - 6x + 8

x = 2 or 4

8

X

V

x+2

x

Z

2x+1

Y

You are at an indoor climbing wall. Using a mirror to see the top of the wall:

CE = 85 ft. AC = 6.5 ft. AB = 5 ft.

How tall is the wall?

The triangles are similar by AA ~ Postulate.

1. Write proportion

2. Substitute given values

3. Solve for DE

D

B

C

A

E

Side-Side-Side (SSS) ~Theorem

If the lengths of the 3 corresponding sides of two triangles are proportional, then the triangles are similar.

Side-Angle-Side (SAS) ~Theorem

If an angle of one triangle is congruent to an angle of a second triangle, and the 2 pairs of sides including these angles are proportional, then the triangles are similar.

CD = ? EC= ? Show that corresponding sides are proportional. Then use SAS Similarity with vertical angles ACB and DCE.

A

6

E

C

9

B

D

Which of these

three triangles

are similar?

1. Compare ratios of side lengths of ΔABC and ΔDEF

shortest sideslongest sidesremaining sides

Because the ratios are all equal, ΔABC ~ ΔDEF.

2. Compare ratios of side lengths of ΔABC and ΔGHJ

shortest sideslongest sidesremaining sides

Because the ratios are not equal, ΔABC and ΔGHJ are not similar.

Since ΔABC~ ΔDEF and ΔABC is not ~ΔGHJ,

ΔDEF is not ~ ΔGHJ.

If corresponding sides of similar polygons are in the ratio a:b, then

- the ratio of the perimeters is a:b.
- the ratio of ANY two corresponding lengths
is a:b (altitudes, medians, angle bisector segments, diagonals, etc.)

- The ratio of the areas is a2:b2

- Write proportion and solve:

42 ft

P

L

M

5 ft

Q

N

8 ft