Question

which congruence theorems can be used to prove δabr ≅ δacr? select three options. □ hl □ sas □ sss □ asa □ aas

Explanation:

Step1: Check HL (Hypotenuse - Leg)

In right - angled triangles \( \triangle ABR\) and \( \triangle ACR\), \(AB = AC\) (given as equal segments), \(AR=AR\) (common hypotenuse). So, by HL (Hypotenuse - Leg) congruence theorem for right - angled triangles, \( \triangle ABR\cong\triangle ACR\).

Step2: Check SSS (Side - Side - Side)

\(AB = AC\), \(BR = CR\) (given as equal segments), \(AR = AR\) (common side). So, by SSS (Side - Side - Side) congruence theorem, \( \triangle ABR\cong\triangle ACR\).

Step3: Check AAS (Angle - Angle - Side)

\(\angle ABR=\angle ACR = 90^{\circ}\), \(AB = AC\), \(AR = AR\). Also, \(\angle BAR=\angle CAR\) (since in right - angled triangles with \(AB = AC\) and \(AR\) as a common side, the angles opposite to the equal sides \(BR\) and \(CR\) are equal). So, by AAS (Angle - Angle - Side) congruence theorem, \( \triangle ABR\cong\triangle ACR\).

Answer:

HL, SSS, AAS