Question

determine whether the triangles must be congruent. if so, name the postulate or theorem that best justifies your answer. if not, explain. choose the correct answer below. a. the triangles are congruent by asa b. the triangles are congruent by sas. c. the triangles are congruent by aas. d. the triangles are congruent by sss. e. the triangles are not congruent.

Explanation:

• In the given figure, we have two right - angled triangles \(\triangle ABD\) and \(\triangle CBD\).
• We know that \(\angle ABD=\angle CBD = 90^{\circ}\) (right angles).
• We also have \(\angle A=\angle C\) (given as angles in the non - right angle positions of the two triangles) and \(BD = BD\) (common side).
• The AAS (Angle - Angle - Side) congruence theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent. Here, the two angles (\(\angle A\) and \(\angle ABD\) in \(\triangle ABD\); \(\angle C\) and \(\angle CBD\) in \(\triangle CBD\)) and the non - included side \(BD\) satisfy the AAS condition.

Answer:

C. The triangles are congruent by AAS.