Question

circle d is shown with the measures of the minor arcs. which angles are congruent?
\\( \angle edh \\) and \\( \angle fdg \\)
\\( \angle fde \\) and \\( \angle gdh \\)
\\( \angle gdh \\) and \\( \angle edh \\)
\\( \angle gdf \\) and \\( \angle hdg \\)

Explanation:

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<reasoning>

Find the measures of the central angles corresponding to the given minor arcs

\[

$$\begin{aligned} &m\angle\text{EDF} = m(\text{arc EF}) = 115^\circ\\ &m\angle\text{FDG} = m(\text{arc FG}) = 115^\circ\\ &m\angle\text{GDH} = m(\text{arc GH}) = 65^\circ\\ &m\angle\text{HDE} = m(\text{arc HE}) = 65^\circ \end{aligned}$$

\]

Identify congruent central angles by comparing their measures

\[

$$\begin{aligned} &m\angle\text{EDF} = m\angle\text{FDG} = 115^\circ \implies \angle\text{FDE} \cong \angle\text{FDG}\\ &m\angle\text{GDH} = m\angle\text{EDH} = 65^\circ \implies \angle\text{GDH} \cong \angle\text{EDH} \end{aligned}$$

\]

Match the congruent pairs with the given multiple-choice options

\[

$$\begin{aligned} &\text{Option 1: } \angle\text{EDH } (65^\circ) \text{ and } \angle\text{FDG } (115^\circ) \quad (\text{Not congruent})\\ &\text{Option 2: } \angle\text{FDE } (115^\circ) \text{ and } \angle\text{GDH } (65^\circ) \quad (\text{Not congruent})\\ &\text{Option 3: } \angle\text{GDH } (65^\circ) \text{ and } \angle\text{EDH } (65^\circ) \quad (\text{Congruent})\\ &\text{Option 4: } \angle\text{GDF } (115^\circ) \text{ and } \angle\text{HDG } (65^\circ) \quad (\text{Not congruent}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>O \(\angle\text{EDH}\) and \(\angle\text{FDG}\)</mcq-option>
<mcq-option>O \(\angle\text{FDE}\) and \(\angle\text{GDH}\)</mcq-option>
<mcq-correct>O \(\angle\text{GDH}\) and \(\angle\text{EDH}\)</mcq-correct>
<mcq-option>O \(\angle\text{GDF}\) and \(\angle\text{HDG}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the measures of the central angles corresponding to the given minor arcs

\[

$$\begin{aligned} &m\angle\text{EDF} = m(\text{arc EF}) = 115^\circ\\ &m\angle\text{FDG} = m(\text{arc FG}) = 115^\circ\\ &m\angle\text{GDH} = m(\text{arc GH}) = 65^\circ\\ &m\angle\text{HDE} = m(\text{arc HE}) = 65^\circ \end{aligned}$$

\]

Identify congruent central angles by comparing their measures

\[

$$\begin{aligned} &m\angle\text{EDF} = m\angle\text{FDG} = 115^\circ \implies \angle\text{FDE} \cong \angle\text{FDG}\\ &m\angle\text{GDH} = m\angle\text{EDH} = 65^\circ \implies \angle\text{GDH} \cong \angle\text{EDH} \end{aligned}$$

\]

Match the congruent pairs with the given multiple-choice options

\[

$$\begin{aligned} &\text{Option 1: } \angle\text{EDH } (65^\circ) \text{ and } \angle\text{FDG } (115^\circ) \quad (\text{Not congruent})\\ &\text{Option 2: } \angle\text{FDE } (115^\circ) \text{ and } \angle\text{GDH } (65^\circ) \quad (\text{Not congruent})\\ &\text{Option 3: } \angle\text{GDH } (65^\circ) \text{ and } \angle\text{EDH } (65^\circ) \quad (\text{Congruent})\\ &\text{Option 4: } \angle\text{GDF } (115^\circ) \text{ and } \angle\text{HDG } (65^\circ) \quad (\text{Not congruent}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>O \(\angle\text{EDH}\) and \(\angle\text{FDG}\)</mcq-option>
<mcq-option>O \(\angle\text{FDE}\) and \(\angle\text{GDH}\)</mcq-option>
<mcq-correct>O \(\angle\text{GDH}\) and \(\angle\text{EDH}\)</mcq-correct>
<mcq-option>O \(\angle\text{GDF}\) and \(\angle\text{HDG}\)</mcq-option>
</answer>

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