Methods of Proving Triangles Similar

Lesson 8.3

Postulate: If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. (AAA)

The following 3 theorems will be used in proofs much as SSS, SAS, ASA, HL and AAS where used in proofs to establish congruency.

Theorem 62: If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar. (AA) (no choice)Theorem 63: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar. (SSS~)

Theorem 64: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar. (SAS~)

Given: ABCD is a

Prove: ∆BFE ~ ∆ CFDA

CD

B

FE

1. ABCD is a 2. AB ║ DC

3. CDF E

4. DFC EFB

5. ∆ BFE ~ ∆CFD

1. Given2. Opposite sides of

a are ║.

3. ║ lines → alt. int. s

4. Vertical s are 5. AA (3, 4)

Given: LP EAN is the midpoint of LP.

P and R trisect EA.

Prove: ∆PEN ~ ∆PAL

AE

N

L

P R

Since LP EA, NPE and LPA are congruent right angles.

If N is the midpoint, of LP, NP = 1 . LP 2But P and R trisect EA so EP = 1 . PA 2Therefore, ∆PEN ~ ∆PAL by SAS ~.