Question

unit exam - congruent triangles
which of the following relationships proves why △acd and △ecb are congruent?
hl
aas
sas
asa

Explanation:

Step1: Analyze given congruent sides

From the diagram, we know \( \overline{BC}\cong\overline{DC} \) and \( \overline{AB}\cong\overline{ED} \). Also, since \( \overline{AC}=\overline{AB}+\overline{BC} \) and \( \overline{EC}=\overline{ED}+\overline{DC} \), by the Segment Addition Postulate and substitution (because \( \overline{AB}\cong\overline{ED} \) and \( \overline{BC}\cong\overline{DC} \)), we get \( \overline{AC}\cong\overline{EC} \).

Step2: Identify the included angle

The angle at \( C \) ( \( \angle ACD \) and \( \angle ECB \)) is common to both triangles \( \triangle ACD \) and \( \triangle ECB \), so \( \angle ACD\cong\angle ECB \) (Reflexive Property of Congruence).

Step3: Apply SAS congruence criterion

We have two sides and the included angle: \( \overline{AC}\cong\overline{EC} \), \( \angle ACD\cong\angle ECB \), and \( \overline{DC}\cong\overline{BC} \). This matches the Side - Angle - Side (SAS) congruence postulate, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. HL (Hypotenuse - Leg) is for right triangles, AAS (Angle - Angle - Side) requires two angles and a non - included side, and ASA (Angle - Side - Angle) requires two angles and the included side. Since we have two sides and the included angle, SAS is the correct criterion.

Answer:

SAS