Question

triangles zfr and tky are shown, where fz ⊥ zr, yk ⊥ tk, fz ≅ tk, and ty ≅ fr. which of the following statements is true?
○ △zfr ≅ △kty by aas congruence
○ △rzf ≅ △ykt by hl congruence
○ △rfz ≅ △ytk by asa congruence
○ there is not enough information to prove the two triangles are congruent

Explanation:

Step1: Recall congruence criteria

AAS (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent.

Step2: Identify angles and sides

Since $FZ\perp ZR$ and $YK\perp TK$, we have $\angle FZR=\angle YKT = 90^{\circ}$. Also, $FZ = TK$ (given) and $TY=FR$ (given). We can assume that vertical angles (if any) or other angle relations (based on the right - angle and side - side - angle - like structure) can be used. In the case of right - triangles, if we consider two angles (the right - angle and another pair of angles, which can be shown equal due to the nature of the figure and given side equalities) and a non - included side (the given equal sides $FZ = TK$ and $TY = FR$), $\triangle ZFR$ and $\triangle KTY$ satisfy AAS.

Step3: Analyze other options

• For HL (Hypotenuse - Leg) in option B, we need to know the hypotenuse. But we are not given information about the hypotenuse lengths.
• For ASA in option A, we need two angles and the included side. But the given information does not match ASA.
• Option D is incorrect as we have enough information (by AAS) to prove congruence.

Answer:

C. $\triangle ZFR\cong\triangle KTY$ by AAS congruence