Question 2

We will use the Side-Side-Side (SSS) congruence criterion to prove the triangles are congruent.

Step 1 — Identify the triangles

Let's consider two such isosceles triangles. These triangles are formed by two radii and a chord. Let the first triangle be $\triangle OAB$. $O$ is the center of the circle. $AB$ is the base (chord). Let the second triangle be $\triangle OPQ$. $PQ$ is the base (chord).

Step 2 — List known equalities

We know $OA$ and $OB$ are radii of the same circle. So, $OA = OB$. We know $OP$ and $OQ$ are radii of the same circle. So, $OP = OQ$. All radii of the same circle are equal in length. Therefore, $OA = OP$. Also, $OB = OQ$. The problem states that the base lengths are equal. So, $AB = PQ$.

Step 3 — Apply SSS congruence criterion

Now, let's compare $\triangle OAB$ and $\triangle OPQ$. We have $OA = OP$ (radii of the same circle). We have $OB = OQ$ (radii of the same circle). We have $AB = PQ$ (given equal base length). By the SSS congruence criterion:

$\triangle OAB \cong \triangle OPQ$

Hence, if two such isosceles triangles have equal base length, they are congruent to each other.

Answer

The two isosceles triangles are congruent by the SSS criterion.