The image shows a triangle $\triangle ABC$, where $\angle A \cong \angle C$. The problem asks which of the following conclusions is true given this information.

Options:

1. $AC \cong BC$
2. $AB \cong BC$
3. $\angle C \cong \angle B$
4. $\angle A \cong \angle B$

Solution:

If $\angle A \cong \angle C$, the triangle $\triangle ABC$ is isosceles with the base $AB$. In an isosceles triangle, the two angles opposite the equal sides are congruent. Therefore, sides $AB$ and $BC$ must be congruent because these sides are opposite the congruent angles $\angle A$ and $\angle C$.

Thus, the correct conclusion is:

• $AB \cong BC$ (Option 2).

Would you like further details or explanations on this?

Here are 5 related questions to explore:

1. How do you identify congruent triangles?
2. What is the relationship between angles and sides in isosceles triangles?
3. How can you prove triangles are congruent using angle-side relationships?
4. How does the Isosceles Triangle Theorem help in solving such problems?
5. What are other common properties of isosceles triangles?

Tip: In an isosceles triangle, the two angles opposite the equal sides are always congruent, and this can often help in solving triangle problems.