Question

1. match the correct answers.

given: ( m angle mno = 50 ^ { circ } ; overline { m p } perp overline { m n } ; overline { o p } perp overline { o n } ; m p = o p )
what is the measure of ( angle m n p ) ?
by which theorem?

Explanation:

Step1: Recall the Angle Bisector Theorem

The Angle Bisector Theorem states that if a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Conversely, if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. Here, since \(MP\perp MN\), \(OP\perp ON\) and \(MP = OP\), point \(P\) lies on the bisector of \(\angle MNO\).

Step2: Calculate the measure of \(\angle MNP\)

Let \(\angle MNP=x\) and \(\angle PNO = x\) (because \(NP\) is the angle - bisector). We know that \(\angle MNO=\angle MNP+\angle PNO\). Given \(\angle MNO = 50^{\circ}\), then \(x + x=50^{\circ}\). So \(2x = 50^{\circ}\), and \(x=\frac{50^{\circ}}{2}=25^{\circ}\)

Answer:

The measure of \(\angle MNP\) is \(25^{\circ}\). By the Angle Bisector Theorem.