Question

find the measure of the arc.

(m\widehat{efd} = \\ ?\\ ^{\circ})

Explanation:

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<reasoning>

Identify the given angles

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &\angle CDE = 90^\circ \quad (\text{indicated by the right-angle symbol})\\ &\angle EDF = 146^\circ \end{aligned}$$

\]

Calculate the remaining central angle

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &\angle FDC = 360^\circ - (\angle CDE + \angle EDF)\\ &\angle FDC = 360^\circ - (90^\circ + 146^\circ) = 124^\circ \end{aligned}$$

\]

Determine the measure of the major arc EFD

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &m\widehat{EFD} = 360^\circ - m\widehat{ED}\\ &m\widehat{EFD} = 360^\circ - 90^\circ = 270^\circ \end{aligned}$$

\]
</reasoning>

<answer>
Find the measure of the arc.

\(m\widehat{EFD} =\) <blank>270</blank>\(^\circ\)
</answer>

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Answer:

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<reasoning>

Identify the given angles

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &\angle CDE = 90^\circ \quad (\text{indicated by the right-angle symbol})\\ &\angle EDF = 146^\circ \end{aligned}$$

\]

Calculate the remaining central angle

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &\angle FDC = 360^\circ - (\angle CDE + \angle EDF)\\ &\angle FDC = 360^\circ - (90^\circ + 146^\circ) = 124^\circ \end{aligned}$$

\]

Determine the measure of the major arc EFD

Using the Central Angle Theorem and Arc Measure knowledge points
\[

$$\begin{aligned} &m\widehat{EFD} = 360^\circ - m\widehat{ED}\\ &m\widehat{EFD} = 360^\circ - 90^\circ = 270^\circ \end{aligned}$$

\]
</reasoning>

<answer>
Find the measure of the arc.

\(m\widehat{EFD} =\) <blank>270</blank>\(^\circ\)
</answer>

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