Question

△jkl is a right triangle, and △jml is an isosceles triangle where \overline{jl} \cong \overline{ml}. which is the measure of ∠klm?

Explanation:

Step1: Find ∠JML

Since △JML is isosceles with \( \overline{JL}\cong\overline{ML} \), and ∠J = 63°, then ∠M = 63° (base angles of an isosceles triangle are equal).

Step2: Find ∠JLM

In △JML, using the angle - sum property of a triangle (\( \angle J+\angle M+\angle JLM = 180^{\circ} \)), we have \( \angle JLM=180^{\circ}-63^{\circ}-63^{\circ}=54^{\circ} \).

Step3: Find ∠KLM

In △JKL, ∠JKL = 90°, ∠JLK = 42°. In the larger context, \( \angle KLM=\angle JLK+\angle JLM \).
Substitute the values: \( \angle KLM = 42^{\circ}+54^{\circ}=96^{\circ} \). Wait, no. Wait, re - check. Wait, no, in △JKL, ∠JKL = 90°, ∠JLK = 42°, so ∠LJK=180 - 90 - 42=48°. Wait, no, wrong approach.

Wait, correct approach:
Since △JML is isosceles (\( JL = ML \)), ∠J = ∠M = 63°, so ∠JLM=180 - 63 - 63 = 54°.
In △JKL, ∠JKL = 90°, ∠JLK = 42°, and we want to find ∠KLM.
We know that \( \angle KLM=\angle JLM+\angle JLK \) (by angle addition).
Substitute the values: \( \angle KLM=54^{\circ}+48^{\circ}=102^{\circ} \).

Answer:

\( 102^{\circ} \)