Question

find the measure of the given angle.
if ( mangle mpn = 32^{circ} ) and ( mangle 3 = 89^{circ} ), find ( mangle rpn ).

Explanation:

Step1: Recall the angle - sum property around a point

The sum of angles around a point \(P\) is \(360^{\circ}\). But if we consider the non - overlapping angles \(\angle MPN\), \(\angle RPN\) and \(\angle3\) (assuming they are part of a full - circle or a relevant angle - sum relationship. If we assume that \(\angle MPN+\angle RPN+\angle3 = 180^{\circ}\) (if they form a linear - pair or a semi - circle like relationship, which is a common geometric relationship when dealing with angles at a point from adjacent rays).
Let \(x = m\angle RPN\).
We know that \(m\angle MPN = 32^{\circ}\) and \(m\angle3=89^{\circ}\).
Using the equation \(m\angle MPN+m\angle RPN+m\angle3 = 180^{\circ}\) (assuming a straight - line or semi - circular angle sum).
Substitute the known values into the equation: \(32^{\circ}+x + 89^{\circ}=180^{\circ}\).

Step2: Solve for \(x\)

First, simplify the left - hand side of the equation: \(32 + 89+x=180\).
\(121+x = 180\).
Subtract \(121\) from both sides of the equation: \(x=180 - 121\).

Answer:

\(59^{\circ}\)