Question

ex4: which congruency theorem can be used to prove that these triangles are congruent?
multiple-choice options: sas, asa, sss, aas, hl

Explanation:

Step1: Analyze triangle properties

The triangles are right - angled (right angle symbol), have one pair of equal legs (single tick mark), and the hypotenuses are equal (double tick marks). Also, the vertical angles formed by the intersecting hypotenuses are equal. But for right - angled triangles, the HL (Hypotenuse - Leg) congruence criterion states that if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, then the triangles are congruent. Here, we have right triangles, equal hypotenuses (double tick) and equal legs (single tick). Also, another way: we can see that we have a right angle, a leg, and the hypotenuse. But also, let's check the options. The HL (Hypotenuse - Leg) is a congruence criterion for right triangles. Let's check the other options:

• SAS: We need two sides and the included angle. Here, the angle between the leg and hypotenuse is not necessarily the included angle in the way SAS requires.
• ASA: We need two angles and the included side. We don't have two angles given in a way that fits ASA.
• SSS: We would need all three sides. We have two sides (leg and hypotenuse) equal, but the third side (the other leg) is equal by Pythagoras, but the criterion HL is more direct for right triangles.
• AAS: We need two angles and a non - included side. Not the case here.
• HL: Since they are right triangles, hypotenuse is equal (double tick), one leg is equal (single tick), so HL applies.

Step2: Confirm the correct option

From the analysis, the HL (Hypotenuse - Leg) congruence theorem is applicable here as the triangles are right - angled, have equal hypotenuses, and equal legs.

Answer:

HL (the option corresponding to HL, with its checkbox selected)