Bot GPT 4-o mini To determine which congruence theorem can be used to prove that triangles \( \triangle WXS \) and \( \triangle YZS \) are congruent, let's analyze the given information:
- \( \angle WXS \) and \( \angle SZY \) are right angles, which means they each measure \( 90^\circ \).
- Sides \( XS \) and \( ZS \) are congruent, i.e., \( XS \cong ZS \).
Now we can outline the necessary parts for the congruence criteria:
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For SAS (Side-Angle-Side): We would need to have two sides and the included angle. Here we have \( XS \cong ZS \) (one side), and \( \angle WXS \cong \angle SZY \) (included angles), but we need another side from \( \triangle WXS \).
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For ASA (Angle-Side-Angle): We would need two angles and the included side to be known. Here, although the angles are both right angles, we do not have the angles at the other vertices defined (i.e., \( \angle WXW \) or \( \angle YSY \)).
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For SSS (Side-Side-Side): We would need three sides of both triangles to be congruent. We only have the length for one pair of sides so this does not apply.
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For HL (Hypotenuse-Leg): This specific theorem applies only to right triangles and states that if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Given that both triangles are right triangles with the hypotenuse (the sides opposite to the right angles, which are \( WX \) and \( YZ \)), and one leg (the sides \( XS \) and \( ZS \)), it appears this theorem may apply.
Since both triangles \( \triangle WXS \) and \( \triangle YZS \) are right triangles with:
- Congruent legs \( XS \cong ZS \)
- Congruent hypotenuses \( WX \) and \( YZ \)
we can conclude that the correct answer is:
HL (Hypotenuse-Leg).
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