Question

which congruence theorems can be used to prove δabr ≅ δacr? select three options. image of triangles: δabr and δacr with right angles at b and c, ab ≅ ac (marked), br ≅ cr (marked), ar is common. options: hl, sas, sss, asa, aas

Explanation:

1. HL (Hypotenuse - Leg): $\triangle ABR$ and $\triangle ACR$ are right - angled triangles ( $\angle ABR=\angle ACR = 90^{\circ}$). We know that $AB = AC$ (marked as equal) and $AR$ is the hypotenuse common to both triangles. So, by HL congruence criterion, $\triangle ABR\cong\triangle ACR$.
2. SAS (Side - Angle - Side): We have $AB = AC$ (given), $\angle ABR=\angle ACR = 90^{\circ}$, and $BR = CR$ (marked as equal). So, two sides and the included angle are equal. By SAS congruence criterion, $\triangle ABR\cong\triangle ACR$.
3. SSS (Side - Side - Side): We know that $AB = AC$, $BR = CR$, and $AR$ is common to both triangles. So, all three corresponding sides are equal. By SSS congruence criterion, $\triangle ABR\cong\triangle ACR$.

For ASA, we do not have two angles and the included side in a way that satisfies the ASA criterion. For AAS, we also do not have two angles and a non - included side in a way that satisfies the AAS criterion.

Answer:

HL, SAS, SSS