Question

circumcenter and incenter open notes quiz
if p is the incenter of \\( \triangle xyz \\), find \\( m\angle z p \\) in the figure below.
possible points: 10
***just type the number and
nothing else.

Explanation:

Step1: Recall the property of the in - center

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle.

Step2: Use the angle sum property of a triangle

In \(\triangle XYZ\), assume the angle bisectors of \(\angle X\), \(\angle Y\), and \(\angle Z\) meet at \(P\).
We know that the sum of angles in a triangle is \(180^{\circ}\). Let \(\angle X = 2\times31^{\circ}=62^{\circ}\) (since \(XP\) is an angle bisector) and \(\angle Y=2\times27^{\circ} = 54^{\circ}\) (since \(YP\) is an angle bisector)
Then \(\angle Z=180^{\circ}-\angle X - \angle Y\)
\(\angle Z=180-(62 + 54)=180 - 116=64^{\circ}\)

Step3: Find \(\angle LZP\)

Since \(ZP\) is an angle bisector of \(\angle Z\), then \(\angle LZP=\frac{1}{2}\angle Z\)
\(\angle LZP=\frac{64^{\circ}}{2}=32^{\circ}\)

Answer:

32