In the figure below, $\angle A P E$ and $\angle E P D$ are congruent.
What is the arc measure of major arc $B \widehat{A} D$ on circle $P$ in degrees?

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the measure of major arc $B \widehat{A} D$, we first need to determine the measure of the central angle $\angle APD$ that intercepts it. ‖ ‖ step 2 ⋮ Since $\angle APE$ and $\angle EPD$ are congruent, we can express $\angle APD$ as the sum of $\angle APE$ and $\angle EPD$. ‖ ‖ step 3 ⋮ The sum of all central angles in a circle is $360^\circ$. We can calculate the measure of $\angle APE$ (and $\angle EPD$) by subtracting the measures of $\angle APB$, $\angle BPC$, and $\angle CPD$ from $360^\circ$ and then dividing by 2, because there are two congruent angles. ‖ ‖ step 4 ⋮ Subtracting the given angles from $360^\circ$: $360^\circ - 136^\circ - 74^\circ - 42^\circ = 108^\circ$. ‖ ‖ step 5 ⋮ Dividing $108^\circ$ by 2 to find the measure of $\angle APE$ and $\angle EPD$: $108^\circ / 2 = 54^\circ$. ‖ ‖ step 6 ⋮ Now, we can find the measure of $\angle APD$ by adding the measures of $\angle APE$ and $\angle EPD$: $54^\circ + 54^\circ = 108^\circ$. ‖ ‖ step 7 ⋮ The measure of major arc $B \widehat{A} D$ is the difference between $360^\circ$ and the measure of $\angle APD$: $360^\circ - 108^\circ = 252^\circ$. ‖ ∻Answer∻ ⚹ The arc measure of major arc $B \widehat{A} D$ on circle $P$ is $252^\circ$. ⚹ ∻Key Concept∻ ⚹Central and Inscribed Angles⚹ ∻Explanation∻ ⚹The measure of a major arc is found by subtracting the central angle that intercepts the minor arc from $360^\circ$. In this case, the major arc is opposite the central angle $\angle APD$.⚹◊What is the formula for calculating the arc measure of a major arc in degrees on a circle?⍭ Generate me a similar question◊

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