Question

1. (3 points) in the diagram below, ( mangle n = 38^{circ}, mangle m = mangle p = 88^{circ} ), and ( mangle o = 54^{circ} ). determine if triangle ( lmn ) and triangle ( opq ) are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Calculate ∠L in △LMN

Sum of angles in triangle is 180°.
$m\angle L = 180^\circ - m\angle M - m\angle N = 180^\circ - 88^\circ - 38^\circ = 54^\circ$

Step2: Calculate ∠Q in △OPQ

Sum of angles in triangle is 180°.
$m\angle Q = 180^\circ - m\angle O - m\angle P = 180^\circ - 54^\circ - 88^\circ = 38^\circ$

Step3: Compare angles of both triangles

△LMN angles: $∠L=54^\circ$, $∠M=88^\circ$, $∠N=38^\circ$
△OPQ angles: $∠O=54^\circ$, $∠P=88^\circ$, $∠Q=38^\circ$
All corresponding angles are equal.

Answer:

Triangles LMN and OPQ are similar by AA (Angle-Angle) similarity criterion (all corresponding angles are equal).