in the diagram below, $\overline{ab}$ and $\overline{bc}$ are tangent to $\odot o$. what is the measure of $\angle abc$?

a. $67^\circ$
b. $46^\circ$
c. $92^\circ$
d. $42^\circ$

Question

in the diagram below, $\overline{ab}$ and $\overline{bc}$ are tangent to $\odot o$. what is the measure of $\angle abc$?

a. $67^\circ$
b. $46^\circ$
c. $92^\circ$
d. $42^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Major Arc", "Minor Arc", "Angles Outside Circle Theorem" ], "new_concepts": [], "current_concepts": [ "Major Arc", "Minor Arc", "Angles Outside Circle Theorem", "Tangent Line" ] } ## Identify the given arc measures Using the *Major Arc* and *Minor Arc* knowledge points $$ \begin{aligned} &m\widehat{ADC} = 226^\circ \quad (\text{Major Arc})\ &m\widehat{AC} = 134^\circ \quad (\text{Minor Arc}) \end{aligned} $$ ## Apply the Angles Outside Circle Theorem Using the *Angles Outside Circle Theorem* knowledge point $$ \begin{aligned} m\angle ABC &= \frac{m\widehat{ADC} - m\widehat{AC}}{2}\ m\angle ABC &= \frac{226^\circ - 134^\circ}{2} \end{aligned} $$ ## Calculate the final angle measure $$ \begin{aligned} m\angle ABC &= \frac{92^\circ}{2}\ m\angle ABC &= 46^\circ \end{aligned} $$

(A) $67^\circ$

(B) $46^\circ$

(C) $92^\circ$

(D) $42^\circ$

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

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QUESTION IMAGE

Question

Explanation:

Identify the given arc measures

Apply the Angles Outside Circle Theorem

Calculate the final angle measure

Answer: