Given that and , show that . Are the two triangles congruent?

Question

Homi · Accepted Answer

Understand the Question

We are given two triangles, $\triangle ABC$ and $\triangle DBC$ , sharing a common side $BC$ .
We are given two pairs of equal angles:
- $\angle ABC = \angle DBC$
- $\angle ACB = \angle DCB$
The shared side $BC$ lies between these two pairs of angles, allowing us to establish congruence using the Angle-Side-Angle (ASA) criterion.
Once congruence is proven, all corresponding parts (angles and sides) of the two triangles must be equal.

Step 1 · Prove Triangle Congruence using ASA Criterion

In $\triangle ABC$ and $\triangle DBC$ : Diagram 1

$\angle ABC = \angle DBC \quad \text{(Given)}$

$BC = BC \quad \text{(Common side)}$

$\angle ACB = \angle DCB \quad \text{(Given)}$

Therefore, by the Angle-Side-Angle (ASA) congruence criterion: $\triangle ABC \cong \triangle DBC$

Step 2 · Show Equal Corresponding Angles

Since $\triangle ABC \cong \triangle DBC$ , their corresponding parts are equal:

$\angle BAC = \angle BDC$

Answer

Yes, $\triangle ABC \cong \triangle DBC$ (by ASA criterion), and $\angle BAC = \angle BDC$ (corresponding parts of congruent triangles).

Common Mistakes

Overlooking the Common Side: Forgetting to identify the shared line segment $BC = BC$ as the included side between the two given angles.
Misidentifying the Criterion: Confusing the ASA criterion with AAS or SAS; here the shared side $BC$ lies strictly between the two pairs of given angles.

Question 12