Question

1. find each angle measure.

a. ( mangle apb )
b. ( mangle cpe )
c. ( mangle bpc )
d. ( mangle apf )
e. ( mangle dpf )

Explanation:

Step1: Find \(m\angle APB\)

From the figure, \(\angle APB = 90^{\circ}\) (right - angle)

Step2: Find \(m\angle CPE\)

\(\angle CPE=\angle CPD+\angle DPE\)
\(\angle CPE = 35^{\circ}+22^{\circ}=57^{\circ}\)

Step3: Find \(m\angle BPC\)

Since \(\angle APB = 90^{\circ}\) and \(\angle APC+\angle BPC = 90^{\circ}\), and \(\angle APC\) and \(\angle DPE\) are vertical angles (\(\angle APC = 22^{\circ}\)), so \(m\angle BPC=90^{\circ}- 22^{\circ}=68^{\circ}\)

Step4: Find \(m\angle APF\)

\(\angle APF\) and \(\angle APC\) are vertical angles. \(\angle APC = 22^{\circ}\), so \(m\angle APF = 22^{\circ}\)

Step5: Find \(m\angle DPF\)

\(\angle DPF=180^{\circ}-\angle DPC\)
Since \(\angle DPC = 35^{\circ}\), then \(m\angle DPF=180^{\circ}-35^{\circ}=145^{\circ}\)

Answer:

a. \(90^{\circ}\)
b. \(57^{\circ}\)
c. \(68^{\circ}\)
d. \(22^{\circ}\)
e. \(145^{\circ}\)