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 the third angle of triangle \(LMN\)

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle LMN\), if \(m\angle N = 38^{\circ}\) and \(m\angle M=88^{\circ}\), then \(m\angle L=180^{\circ}-(38^{\circ} + 88^{\circ})\).

$$m\angle L=180^{\circ}-126^{\circ}=54^{\circ}$$

Step2: Check the similarity using the AA (Angle - Angle) similarity criterion

In \(\triangle LMN\), the angles are \(m\angle L = 54^{\circ}\), \(m\angle M = 88^{\circ}\), \(m\angle N=38^{\circ}\). In \(\triangle OPQ\), the angles are \(m\angle O = 54^{\circ}\), \(m\angle P = 88^{\circ}\). Since two pairs of corresponding angles are equal (\(\angle L\cong\angle O = 54^{\circ}\) and \(\angle M\cong\angle P=88^{\circ}\)), by the AA (Angle - Angle) similarity criterion.

Answer:

Triangles \(LMN\) and \(OPQ\) are similar by the AA (Angle - Angle) similarity criterion.