Question

quiz 3: similar triangle
a
m
n
b
c
given ∠amn = 45° and ∠abc = 45°, the triangles are similar by

the sss similarity theorem.

the sas similarity theorem.

the aa similarity postulate.

the triangles are not similar.

Explanation:

Step1: Analyze given angles

We know that $\angle AMN = 45^\circ$ and $\angle ABC = 45^\circ$. Also, $\angle A$ is common to both $\triangle AMN$ and $\triangle ABC$.

Step2: Identify similarity criterion

The AA (Angle - Angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Here, $\angle AMN\cong\angle ABC$ (both $45^\circ$) and $\angle A\cong\angle A$ (common angle), so by AA similarity postulate, the triangles are similar. The SSS similarity theorem requires three pairs of proportional sides, and SAS requires two sides in proportion and the included angle congruent, which is not given here. Also, the triangles are similar, so the last option is incorrect.

Answer:

the AA similarity postulate.