Question

k extra practice level up your skills! mixed proofs find the measure of each angle. 1. find ( m angle r p n ) if ( m angle m p n = 35^{circ} ) and ( m angle 3 = 90^{circ} ). 2. find ( m angle l p n ) if ( angle 3 ) is a right angle and ( angle m p n cong angle r p n ).

Explanation:

Step1: Use the property of angles around a point

Since \(m\angle3 = 90^{\circ}\), and \(\angle MPN+\angle RPN=m\angle3\) (because \(\angle MPN\) and \(\angle RPN\) are adjacent angles that form \(\angle3\)).
Given \(m\angle MPN = 35^{\circ}\), then \(m\angle RPN=m\angle3 - m\angle MPN\).

Step2: Substitute the values

Substitute \(m\angle3 = 90^{\circ}\) and \(m\angle MPN = 35^{\circ}\) into the formula \(m\angle RPN=m\angle3 - m\angle MPN\).
We get \(m\angle RPN=90^{\circ}- 35^{\circ}\).

Step3: Calculate the result

\(90^{\circ}-35^{\circ}=55^{\circ}\).

Step4: For the second part

Since \(\angle3\) is a right - angle (\(m\angle3 = 90^{\circ}\)) and \(\angle MPN\cong\angle RPN\), let \(m\angle MPN = m\angle RPN=x\). Then \(x + x=90^{\circ}\) (because \(\angle MPN+\angle RPN=\angle3\)).
Solving \(2x = 90^{\circ}\), we get \(x = 45^{\circ}\).
Also, \(\angle LPM\) and \(\angle3\) are vertical angles, so \(m\angle LPM=m\angle3 = 90^{\circ}\).
And \(m\angle LPN=m\angle LPM + m\angle MPN\).
Substitute \(m\angle LPM = 90^{\circ}\) and \(m\angle MPN = 45^{\circ}\), we get \(m\angle LPN=90^{\circ}+45^{\circ}=135^{\circ}\).

Answer:

1. \(m\angle RPN = 55^{\circ}\)
2. \(m\angle LPN=135^{\circ}\)