Question

If is an isosceles triangle and D is the midpoint of

, what reason can be used to prove that the triangles are congruent?

Answer 1

To prove that two triangles are congruent, you can use various congruence criteria. Given that you have an isosceles triangle where \(D\) is the midpoint of the base \(BC\), here are the steps showing how to prove that the two triangles \(ABD\) and \(ACD\) are congruent:

1. **Definition of Isosceles Triangle**: Since the triangle \(ABC\) is isosceles, we have \(AB = AC\) (the two sides opposite the base \(BC\) are equal).

2. **Midpoint Definition**: Point \(D\) is given as the midpoint of \(BC\), which means \(BD = DC\).

3. **Shared Side**: The segments \(AD\) are shared by both triangles \(ABD\) and \(ACD\), so \(AD = AD\).

With this information, we have:

- \(AB = AC\) (two sides of the isosceles triangle),
- \(BD = DC\) (since \(D\) is the midpoint),
- \(AD = AD\) (the same segment for both triangles).

Using the **Side-Side-Side (SSS)** or **Side-Angle-Side (SAS)** congruence criterion, you can conclude that triangles \(ABD\) and \(ACD\) are congruent. Specifically, you can apply SAS:

- \(AB = AC\) (two sides),
- \(AD = AD\) (common side),
- \(BD = DC\) (two sides).

Therefore, you can state that triangles \(ABD\) and \(ACD\) are congruent by the SAS congruence criterion.