in circle d, angle adc measures $(7x + 2)^\circ$. arc ac measures $(8x - 8)^\circ$.

what is the measure of $\angle abc$?

$36^\circ$
$43^\circ$
$72^\circ$
$144^\circ$

Question

in circle d, angle adc measures $(7x + 2)^\circ$. arc ac measures $(8x - 8)^\circ$.

what is the measure of $\angle abc$?

$36^\circ$
$43^\circ$
$72^\circ$
$144^\circ$

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", "Inscribed Angle Theorem" ], "new_concepts": [], "current_concepts": [ "Central Angle Theorem", "Inscribed Angle Theorem", "Circle Geometry" ] } ## Equate the central angle and its intercepted arc $$ \begin{aligned} m\angle ADC &= m(\text{Arc } AC) \ 7x + 2 &= 8x - 8 \ x &= 10 \end{aligned} $$ ## Calculate the measure of the central angle and arc $$ \begin{aligned} m\angle ADC &= 7(10) + 2 = 72^\circ \ m(\text{Arc } AC) &= 72^\circ \end{aligned} $$ ## Apply the Inscribed Angle Theorem to find the measure of angle ABC $$ \begin{aligned} m\angle ABC &= \frac{1}{2} \cdot m(\text{Arc } AC) \ m\angle ABC &= \frac{1}{2} \cdot 72^\circ = 36^\circ \end{aligned} $$

(A) 36°

(B) 43°

(C) 72°

(D) 144°

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

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QUESTION IMAGE

Question

Explanation:

Equate the central angle and its intercepted arc

Calculate the measure of the central angle and arc

Apply the Inscribed Angle Theorem to find the measure of angle ABC

Answer: