Methods of Proving Triangles Similar

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Methods of Proving Triangles Similar

• Lesson 8.3

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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.
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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)
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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)
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Given ABCD is a Prove ?BFE ? CFD
D
C
F
E
A
B

1. ABCD is a
2. AB DC
3. ?CDF ? ?E
4. ?DFC ? ?EFB
5. ? BFE ?CFD

1. Given
2. Opposite sides of a are .
3. lines ? alt. int. ?s ?
4. Vertical ?s are ?
5. AA (3, 4)

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L
Given LP ? EAN is the midpoint of LP. P and R
trisect EA. Prove ?PEN ?PAL
N
A
E
P
R
Since LP ? EA, ?NPE and ?LPA are congruent right
angles. If N is the midpoint, of LP, NP 1
.
LP 2 But P and R trisect EA so EP 1 .
PA
2 Therefore, ?PEN ?PAL by SAS .