Question

Math

Posted 6 months ago

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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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Answer from Sia

Posted 6 months ago

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 $\angle APD$.

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