THEOREM 3 (SAS Similarity Criterion) If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.

To solve this question and provide a detailed answer, let us analyze and explain the SAS Similarity Criterion for triangles step by step:

Given Statement:
If in two triangles,

One pair of corresponding sides are proportional, and
The included angles are equal,
then the two triangles are similar.
To Prove:
Under the given conditions, the two triangles are similar by the SAS (Side-Angle-Side) Similarity Criterion.

Explanation:
Understanding the SAS Similarity Criterion:

For two triangles to be similar, their corresponding angles must be equal, and their corresponding sides must be proportional.

The SAS Similarity Criterion specifically requires:
a. A pair of corresponding sides to have the same ratio (proportional sides).
b. The angle included between these sides in both triangles to be equal.

If these conditions are satisfied, the two triangles are similar.

What is Triangle Similarity?

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
This means the shape of the two triangles is the same, but their sizes may differ (scaled up or down).
Proof:
Let us consider two triangles:

Triangle ABC and Triangle DEF.
Given:

AB/DE = AC/DF (Proportional corresponding sides).
∠A = ∠D (Included angles are equal).
To Prove:
The triangles ABC and DEF are similar.

Steps:

From the given, we know:

AB/DE = AC/DF → This is the condition of proportional sides.
∠A = ∠D → The included angles are equal.
Place the triangles side by side so that:

Side AB in triangle ABC corresponds to side DE in triangle DEF.
Side AC in triangle ABC corresponds to side DF in triangle DEF.
Angle ∠A in triangle ABC corresponds to angle ∠D in triangle DEF.
Now, observe the triangles:

Since one pair of corresponding sides (AB/DE) and another pair of corresponding sides (AC/DF) are proportional, and the angle included between these sides (∠A = ∠D) is equal, we can apply the SAS Similarity Criterion.
By the SAS Similarity Criterion, the two triangles ABC and DEF are similar.

Conclusion:
When one pair of corresponding sides are proportional, and the included angles are equal, the two triangles are similar by the SAS Similarity Criterion.

This satisfies the conditions of triangle similarity, proving the theorem.