Question

find the measure of the given angle.
find ( mangle lpn ) if ( angle 3 ) is a right angle and ( angle mpncongangle rpn ).

Explanation:

Step1: Determine the measure of ∠MPN + ∠RPN

Since ∠3 is a right - angle (\(m\angle3 = 90^{\circ}\)), and the sum of angles around a point is \(360^{\circ}\).
Let \(x=m\angle MPN\) and \(y = m\angle RPN\). We know that \(m\angle LPN+x + y+m\angle3=360^{\circ}\). Also, because \(\angle MPN\cong\angle RPN\), so \(x = y\).
\(m\angle LPN+x + y+90^{\circ}=360^{\circ}\), then \(m\angle LPN+(x + y)=270^{\circ}\).

Step2: Use the property of congruent angles

Since \(x = y\) and \(x + y=270^{\circ}-m\angle LPN\). But we can also use another approach.
The sum of angles around point \(P\) is \(360^{\circ}\). Let \(m\angle MPN=m\angle RPN = z\).
We know that \(m\angle LPN+2z + 90^{\circ}=360^{\circ}\). Also, note that \(\angle LPN\) and \((\angle MPN+\angle RPN+\angle3)\) are supplementary in a sense of the full - circle angle.
Another way: The sum of angles around \(P\) is \(360^{\circ}\).
\(m\angle LPN=360^{\circ}- 90^{\circ}-2m\angle MPN\). But since \(\angle MPN\) and \(\angle RPN\) are congruent and \(\angle3 = 90^{\circ}\).
We know that \(\angle LPN\) and \((\angle MPN+\angle RPN+\angle3)\) form a full - circle.
Since \(\angle MPN\cong\angle RPN\), and the sum of angles around \(P\) is \(360^{\circ}\).
\(m\angle LPN = 180^{\circ}\) (because \(\angle3 = 90^{\circ}\) and \(\angle MPN+\angle RPN=90^{\circ}\) (since \(\angle MPN\cong\angle RPN\), \(m\angle MPN=m\angle RPN = 45^{\circ}\) and \(90^{\circ}+90^{\circ}=180^{\circ}\))

Answer:

\(180\)