Question

two sides and the non-included right angle of one right triangle are congruent to the corresponding parts of another right triangle. which congruence theorem can be used to prove that the triangles are congruent?
○ aas
○ sss
○ sas
○ hl

Explanation:

1. Recall the congruence theorems:

• AAS (Angle - Angle - Side): If two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the triangles are congruent. In right triangles, we know that the right angles are congruent. If we have two sides and a non - included right angle, we can think of it as having a right angle (one angle), another angle (since the sum of angles in a triangle is \(180^{\circ}\) and we know a right angle, the other acute angles are related), and a side. But more directly, for right triangles, when we have two sides and a non - included right angle, we can also use AAS. Let's analyze the other options:
• SSS (Side - Side - Side): Requires all three sides to be congruent. The problem states two sides and a non - included right angle, not three sides, so SSS is out.
• SAS (Side - Angle - Side): Requires the angle to be the included angle between the two sides. The problem states a non - included right angle, so SAS is not applicable.
• HL (Hypotenuse - Leg): Applies to right triangles and requires the hypotenuse and one leg of one right triangle to be congruent to the hypotenuse and one leg of another right triangle. The problem does not specify hypotenuse and leg, but two sides and a non - included right angle.
• For right triangles, we know that \(\angle C=\angle C' = 90^{\circ}\) (right angles). Suppose we have \(AB = A'B'\) and \(BC=B'C'\) and \(\angle C=\angle C' = 90^{\circ}\) (non - included angle for the sides \(AB\) and \(BC\) in the right triangle). We can also consider the angles: in right triangle \(ABC\), \(\angle A=90^{\circ}-\angle B\) and in right triangle \(A'B'C'\), \(\angle A' = 90^{\circ}-\angle B'\). If we have two sides and a non - included right angle, we can use AAS. For example, if we have a right angle, one side, and then we can find the other angle (since the triangle is right - angled) and then use AAS (two angles and a non - included side).

Answer:

A. AAS