Question

could △jkl be congruent to △xyz? 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.

Explanation:

Step1: Recall the Hypotenuse - Leg (HL) Congruence Theorem

For right - angled triangles, if the hypotenuse and one leg of one right - angled triangle are congruent to the hypotenuse and one leg of another right - angled triangle, then the two right - angled triangles are congruent.
In \(\triangle JKL\) and \(\triangle XYZ\), \(\angle J=\angle X = 90^{\circ}\), \(KL = YZ=10\). If \(JL\cong XZ\) (i.e., \(JL = XZ\)), by the HL theorem, \(\triangle JKL\cong\triangle XYZ\).

Step2: Analyze the other options

• If \(XZ = 10\), we don't know the relationship between the hypotenuse and the leg in terms of congruence for the HL theorem. Just knowing \(XZ = 10\) (where \(KL = 10\)) is not sufficient as we need to match hypotenuse - leg pairs.
• The statement “No, because the hypotenuse of one triangle is equal in length to the leg of the other triangle” is incorrect. In our case, if \(JL\cong XZ\), we have a proper hypotenuse - leg match.
• The statement “No, because the leg of one triangle is equal in length to the leg of the other triangle” is also incorrect. We have a right - angled triangle congruence theorem (HL) that can be applied when we have the right angle, a leg, and the hypotenuse.

Answer:

Yes, if \(\overline{JL}\cong\overline{XZ}\) (First option).