Question

given: \\(\overline{ar}\\) bisects \\(\angle bac\\); \\(ab = ac\\)
which congruence theorem can be used to prove \\(\triangle abr \cong \triangle acr\\)?
\\(\bigcirc\\) aas
\\(\bigcirc\\) sss
\\(\bigcirc\\) asa
\\(\bigcirc\\) sas

Explanation:

Step1: Analyze the given information

We know that \(AB = AC\) (given side), \(\overline{AR}\) bisects \(\angle BAC\), so \(\angle BAR=\angle CAR\) (angle), and \(AR = AR\) (common side).

Step2: Recall the congruence theorems

• AAS (Angle - Angle - Side): Requires two angles and a non - included side.
• SSS (Side - Side - Side): Requires three sides.
• ASA (Angle - Side - Angle): Requires two angles and the included side.
• SAS (Side - Angle - Side): Requires two sides and the included angle.

Step3: Match with the SAS theorem

We have two sides (\(AB = AC\) and \(AR=AR\)) and the included angle (\(\angle BAR=\angle CAR\)) between them. So, the SAS (Side - Angle - Side) congruence theorem can be used to prove \(\triangle ABR\cong\triangle ACR\).

Answer:

D. SAS