Could ΔJKL be congruent to ΔXYZ? Explain. Yes, if JL ≅ XZ. Yes, if XZ = 10. No, because the hypotenuse of one triangle is equal in length to the leg of the other triangle. No, because the leg of one triangle is equal in length to the leg of the other triangle.
The Correct Answer and Explanation is:
The correct answer is:
No, because the hypotenuse of one triangle is equal in length to the leg of the other triangle.
Explanation
To determine whether two triangles, ΔJKL and ΔXYZ, are congruent, we must check whether their corresponding sides and angles are exactly equal. Congruent triangles have the same size and shape, meaning all three sides and all three angles must match.
There are several triangle congruence postulates, such as:
- SSS (Side-Side-Side): All three corresponding sides are equal.
- SAS (Side-Angle-Side): Two sides and the included angle are equal.
- ASA (Angle-Side-Angle): Two angles and the included side are equal.
- AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
- HL (Hypotenuse-Leg): For right triangles, the hypotenuse and one leg are equal.
In the context of the question, we are told that the hypotenuse of one triangle is equal in length to a leg of the other triangle. This is a crucial detail. The hypotenuse is always the longest side in a right triangle. A leg is one of the two shorter sides. So if a hypotenuse in ΔJKL is equal in length to a leg in ΔXYZ, this suggests the triangles do not have matching corresponding sides.
Since triangle congruence depends on corresponding parts being equal, a hypotenuse cannot correspond to a leg and still meet the criteria for congruence. For instance, if JL (the hypotenuse of ΔJKL) is equal to XZ (a leg of ΔXYZ), this mismatch in roles means the triangles differ in shape, and thus they cannot be congruent.
Therefore, even if some sides are equal in length, congruence requires those sides to be in corresponding positions. When the hypotenuse of one triangle matches the length of a leg of another, the triangles are not congruent.
