bcd is a circumscribed angle of circle a. angle bac measures 53^

what is the measure of angle bcd?

37^
53^
74^
106^

Question

bcd is a circumscribed angle of circle a. angle bac measures 53^

what is the measure of angle bcd?

37^
53^
74^
106^

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Circumscribed Angle Theorem" ], "new_concepts": [], "current_concepts": [ "Circumscribed Angle Theorem", "Circle Geometry" ] } ## Identify the given angle and geometric relationship $$ \begin{aligned} &\angle BAD = 53^\circ \quad (\text{Central angle intercepting arc } BD) \ &\angle BCD \quad (\text{Circumscribed angle of circle } A) \end{aligned} $$ ## Apply the Circumscribed Angle Theorem $$ \begin{aligned} &\angle BCD + \angle BAD = 180^\circ \ &\angle BCD + 53^\circ = 180^\circ \end{aligned} $$ ## Solve for the measure of angle BCD $$ \begin{aligned} &\angle BCD = 180^\circ - 53^\circ \ &\angle BCD = 127^\circ \end{aligned} $$ *(Note: Re-evaluating the given angle in the diagram. The angle labeled $53^\circ$ is $\angle BAC$, which is half of the central angle $\angle BAD$. Since line $AC$ bisects $\angle BAD$ and $\angle BCD$):* $$ \begin{aligned} &\angle BAC = 53^\circ \ &\angle BCA = 90^\circ - 53^\circ = 37^\circ \ &\angle BCD = 2 \times \angle BCA = 2 \times 37^\circ = 74^\circ \end{aligned} $$

37°

53°

74°

106°

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circumscribed Angle Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given angle and geometric relationship

Apply the Circumscribed Angle Theorem

Solve for the measure of angle BCD

Answer: