Question

let p be the intersection of the angle bisectors of \\( \triangle xyz \\). what is the measurement of \\( \angle lzp \\)?

Explanation:

Step1: Recall the property of angle bisectors

Since \(P\) is the intersection of the angle bisectors of \(\triangle XYZ\), \(XP\), \(YP\), and \(ZP\) are angle bisectors.

Step2: Use the angle - sum property of a triangle

In \(\triangle XYZ\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let's first find \(\angle XZY\).
We know that \(\angle XYZ = 2\times27^{\circ}=54^{\circ}\) (because \(YP\) is the angle bisector) and \(\angle YXZ=2\times31^{\circ} = 62^{\circ}\) (because \(XP\) is the angle bisector).
Using the formula \(\angle XZY=180^{\circ}-\angle YXZ - \angle XYZ\).
Substitute the values: \(\angle XZY=180^{\circ}-62^{\circ}-54^{\circ}=64^{\circ}\).

Step3: Find \(\angle LZP\)

Since \(ZP\) is the angle bisector of \(\angle XZY\), then \(\angle LZP=\frac{1}{2}\angle XZY\).
Substitute \(\angle XZY = 64^{\circ}\), we get \(\angle LZP=\frac{64^{\circ}}{2}=32^{\circ}\).

Answer:

\(32^{\circ}\)