point o is the incenter of triangle abc.
what is $m\angle qob$?
$30^\circ$
$60^\circ$
$75^\circ$
$90^\circ$

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Incenter Theorem", "Angle Bisector" ], "new_concepts": [], "current_concepts": [ "Incenter Theorem", "Angle Bisector", "Right Triangle Trigonometry", "Angle Sum Theorem" ] } ## Identify angle bisector properties of the incenter $$ \begin{aligned} &\text{Since } O \text{ is the incenter of } \triangle ABC, \text{ the segments } BO \text{ and } CO \text{ are angle bisectors.}\ &m\angle OBC = 15^\circ \implies m\angle ABC = 2 \times 15^\circ = 30^\circ\ &m\angle OCB = 30^\circ \implies m\angle ACB = 2 \times 30^\circ = 60^\circ \end{aligned} $$ ## Determine the angles of right triangle OQB $$ \begin{aligned} &OQ \perp AB \implies m\angle OQB = 90^\circ\ &m\angle QBO = m\angle ABO = \frac{1}{2} m\angle ABC = 15^\circ\ &m\angle QOB = 180^\circ - m\angle OQB - m\angle QBO = 180^\circ - 90^\circ - 15^\circ = 75^\circ \end{aligned} $$

30°

60°

75°

90°

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

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Question

Explanation:

Identify angle bisector properties of the incenter

Determine the angles of right triangle OQB

Answer: