Question

1. determine which method if any can prove the triangles are congruent. there is more than one, check all that apply

4.
sss
sas
asa
aas
hl

Explanation:

Step1: Analyze the given information

We have two right - angled triangles (the perpendicular gives the right angle). Let's assume the two triangles are \(\triangle ABD\) and \(\triangle ACD\). We know that \(AB = AC\) (marked as equal), \(\angle ADB=\angle ADC = 90^{\circ}\), and \(AD\) is a common side.

Step2: Check for AAS

In \(\triangle ABD\) and \(\triangle ACD\), we have \(\angle B=\angle C\) (given as equal angles), \(\angle ADB=\angle ADC\) (both right angles), and \(AB = AC\). By the Angle - Angle - Side (AAS) congruence criterion (\(AAS\): if two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent), \(\triangle ABD\cong\triangle ACD\).

Step3: Check for HL

For right - angled triangles (\(\triangle ABD\) and \(\triangle ACD\)), the hypotenuse \(AB = AC\) and the leg \(AD=AD\). By the Hypotenuse - Leg (\(HL\)) congruence criterion (\(HL\): if the hypotenuse and a leg of a right - angled triangle are equal to the hypotenuse and a leg of another right - angled triangle, the two right - angled triangles are congruent), \(\triangle ABD\cong\triangle ACD\).

Answer:

AAS, HL