Question

Given:
∠
A
E
D
≅
∠
A
C
B
.
∠AED≅∠ACB.

Prove:
∠
A
B
C
≅
∠
A
D
E
∠ABC≅∠ADE.

Note: quadrilateral properties are not permitted in this proof.

Answer 1

The ΔAED and ΔBAC share a common vertex angle at point A, and given

that ∠AED ≅ ∠ACB, the two triangles are similar.

The two column proof is presented as follows;

Statement ${}$                       Reason

∠AED ≅ ∠ACB ${}$               Given

∠EAD ≅ ∠CAB  ${}$              Reflexive property

ΔAED ~ ΔBAC  ${}$              AA, Angle-Angle similarity postulate

∠ABC ≅ ∠ADE  ${}$             CASTC

Reasons:

AA Angle-Angle similarity postulate;

The Angle-Angle similarity postulate states that two triangles are similar, if

two angles on one triangle are congruent to two angles on the other

triangle.

Given that two angles on ΔAED are congruent to two angles on ΔBAC, the two triangles are similar, and therefore, the three angles of both triangles are congruent.

CASTC is the acronym for Corresponding Angles of Similar Triangles are Congruent, which points out that given two similar triangles, the angles on one triangle are congruent to the corresponding angle on the other triangle.

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