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7.3 Proving Triangles Similar using AA, SSS, SAS. Angle-Angle (AA) ~ Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. W.  W is reflexive and both triangles have a right  . Δ WVX ~ Δ WZY by AA ~ Post. X.

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7 3 proving triangles similar using aa sss sas

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


Finding distances indirectly
Finding Distances Indirectly

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.


Ac 6 ad 10 bc 9 be 15 describe how to prove that acb dce
AC= 6, AD = 10, BC = 9, BE= 15. Describe how to prove that ΔACB ~ ΔDCE.

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


Using the sss similarity theorem
Using the SSS Similarity Theorem

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.


More about scale factors
More About Scale Factors

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


Use the given lengths to find the width lm of the river below
Use the given lengths to find the width LM of the river below.

  • Write proportion and solve:

42 ft

P

L

M

5 ft

Q

N

8 ft


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