Questions: Quiz: Proving Right Triangle Congruence Based only on the information given in the diagram, which congruence theorems or postulates could be given as reasons why triangle CDE is congruent to triangle OPQ? Check all that apply. A. LL B. SAS C. HL D. AAS E. ASA F. HA

Solution Steps

The diagram shows two right triangles, $\triangle CDE$ and $\triangle OPQ$. We are given that $\overline{CE} \cong \overline{OQ}$. Also, since they are right triangles, $\angle E \cong \angle Q$.

We are looking for congruence theorems that can be applied using the given information. Since we have a right triangle, we can use HL (Hypotenuse-Leg). We know the hypotenuses ($CE$ and $OQ$) are congruent and we have two right angles, but we don't know if the legs are congruent. So we can use HA, given we have a hypotenuse and an acute angle congruent. We can also use LL, given two legs congruent, but we only know one leg is congruent. We can use SAS, if two sides are congruent and their included angle is congruent. AAS if two angles and a side not included are congruent. ASA if two angles and the included side is congruent.

• HL (Hypotenuse-Leg): We have congruent hypotenuses ($CE \cong OQ$), but we do _not_ have congruent legs. Therefore, HL does _not_ apply.
• HA (Hypotenuse-Angle): We have congruent hypotenuses ($CE \cong OQ$), and congruent acute angles ($\angle D \cong \angle P$), so HA applies.
• LL: We do not have enough information. LL requires two pairs of congruent legs, while we only have congruent hypotenuses.
• SAS: We only know one pair of congruent sides so far, so SAS doesn't apply.
• AAS: We have a right angle congruent, however, we only have one other congruent side (hypotenuse). Therefore, we need another congruent angle that is not the right angle, and we don't know if we have that, so AAS doesn't apply.
• ASA: We have the hypotenuses congruent and we have the right angles congruent, which can be used for ASA.

Final Answer: The correct options are HA and ASA.