in the figure below, $\\overline{bd}$ and $\\overline{ec}$ are diameters of circle $p$.\n\nwhat is the arc…

Explanation:

Step1: Identify the central angle relationship

The measure of a minor arc is equal to the measure of its corresponding central angle. Thus, $m\widehat{AE} = m\angle APE$.

Step2: Use the property of vertical angles

Since $\overline{BD}$ and $\overline{EC}$ are diameters intersecting at $P$, $\angle DPE$ and $\angle BPC$ are vertical angles. Therefore, $m\angle BPC = m\angle DPE = 63^{\circ}$.

Step3: Identify the right angle

The figure indicates that $\angle APB$ is a right angle, so $m\angle APB = 90^{\circ}$.

Step4: Calculate the measure of angle APE

Points $E, P, C$ lie on a straight line (diameter). The sum of angles on one side of the line is $180^{\circ}$. $$m\angle APE + m\angle APB + m\angle BPC = 180^{\circ}$$

Step5: Substitute known values and solve

Substitute $90^{\circ}$ and $63^{\circ}$ into the equation: $$m\angle APE + 90^{\circ} + 63^{\circ} = 180^{\circ}$$ $$m\angle APE + 153^{\circ} = 180^{\circ}$$ $$m\angle APE = 180^{\circ} - 153^{\circ} = 27^{\circ}$$

Answer:

27^{\circ}