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 $

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Central Angle Theorem", "Congruent Arcs" ], "new_concepts": [], "current_concepts": [ "Central Angle Theorem", "Congruent Arcs", "Circle Geometry" ] } ## 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 $$ \begin{aligned} &\text{Option 1: } \angle\text{EDH } (65^\circ) \text{ and } \angle\text{FDG } (115^\circ) \quad (\text{Not congruent})\ &\text{Option 2: } \angle\text{FDE } (115^\circ) \text{ and } \angle\text{GDH } (65^\circ) \quad (\text{Not congruent})\ &\text{Option 3: } \angle\text{GDH } (65^\circ) \text{ and } \angle\text{EDH } (65^\circ) \quad (\text{Congruent})\ &\text{Option 4: } \angle\text{GDF } (115^\circ) \text{ and } \angle\text{HDG } (65^\circ) \quad (\text{Not congruent}) \end{aligned} $$

O $\angle\text{EDH}$ and $\angle\text{FDG}$

O $\angle\text{FDE}$ and $\angle\text{GDH}$

O $\angle\text{GDH}$ and $\angle\text{EDH}$

O $\angle\text{GDF}$ and $\angle\text{HDG}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Geometry" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

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

Identify congruent central angles by comparing their measures

Match the congruent pairs with the given multiple-choice options

Answer: