Question

How to prove two triangles are congruent using ASA criterion

Answer 1

To prove that two triangles are congruent using the Angle-Side-Angle (ASA) criterion, you need to follow these steps:

### ASA Criterion
The ASA criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

### Steps to Prove Congruence Using ASA

1. **Identify the Triangles**: Clearly label the two triangles you want to prove congruent, say, \( \triangle ABC \) and \( \triangle DEF \).

2. **Show that Two Pairs of Angles are Congruent**: Establish that:
- \( \angle A \cong \angle D \)
- \( \angle B \cong \angle E \)

3. **Show that the Included Side is Congruent**: Identify and show that the side between the two pairs of angles is equal:
- \( AB \cong DE \) (where \( AB \) is the side between \( \angle A \) and \( \angle B \), and \( DE \) is the side between \( \angle D \) and \( \angle E \))

4. **Apply the ASA Criterion**: Since you have found:
- \( \angle A \cong \angle D \)
- \( \angle B \cong \angle E \)
- \( AB \cong DE \)
You can conclude that:
- \( \triangle ABC \cong \triangle DEF \) by the ASA criterion.

### Example
Suppose you have two triangles, \( \triangle ABC \) and \( \triangle DEF \):
- Show:
- \( \angle A = 50^\circ \), \( \angle B = 60^\circ \)
- \( \angle D = 50^\circ \), \( \angle E = 60^\circ \)
- The included sides \( AB = 10 \) units and \( DE = 10 \) units.

You conclude that:
- \( \triangle ABC \cong \triangle DEF \) by the ASA criterion since you have two pairs of congruent angles and the side between them is also congruent.

### Conclusion
By following these steps and providing sufficient evidence for each of the congruences, you can effectively use the ASA criterion to demonstrate that two triangles are congruent.