Question

decide whether enough information is given to prove that the triangles are congruent. if so, state the theorem you can use. δabc, δdbc
there is not enough information.
there is enough information to use the aas congruence theorem.
there is enough information to use the asa congruence theorem.
there is enough information to use the sas congruence theorem.

Explanation:

Step1: Identify congruent parts

We have two right - angled triangles \(\triangle ABC\) and \(\triangle DBC\). \(\angle ACB=\angle DCB = 90^{\circ}\), \(\angle A=\angle D\) (given as marked angles), and \(BC = BC\) (common side).

Step2: Recall congruence theorems

The AAS (Angle - Angle - Side) congruence theorem states that if two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Step3: Apply the theorem

In \(\triangle ABC\) and \(\triangle DBC\), we have two angles (\(\angle A=\angle D\) and \(\angle ACB=\angle DCB\)) and a non - included side (\(BC\)) equal.

Answer:

There is enough information to use the AAS congruence Theorem.