Question

which reason proves that △jkl is similar to △pqr? two triangles with angles and side lengths: △jkl has ∠j=52°, ∠k=45°, side jk=6, jl=4; △pqr has ∠p=52°, side pq=8. options: a) m∠r=83° b) m∠r=45° c) pr=3/16 d) qr=16/3

Explanation:

Step1: Find angles in △JKL

In △JKL, sum of angles is \(180^\circ\). So \(m\angle L = 180^\circ - 52^\circ - 45^\circ = 83^\circ\).

Step2: Analyze similarity condition

For △JKL and △PQR, \(\angle J = \angle P = 52^\circ\). If \(\angle R = 83^\circ\), then \(\angle Q = 180^\circ - 52^\circ - 83^\circ = 45^\circ\), so all corresponding angles are equal (AA similarity). Check options: A says \(m\angle R = 83^\circ\), which matches \(\angle L\), so AA similarity holds. B is wrong (∠R≠45°). C and D are about side ratios, but AA is angle - based and more direct here.

Answer:

A) \(m\angle R = 83^\circ\)