Question

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

Explanation:

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"Angle Bisector"
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"Angle Sum Theorem"
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<reasoning>

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

\[

$$\begin{aligned} &OQ \perp AB \implies m\angle OQB = 90^\circ\\ &m\angle QBO = m\angle ABO = \frac{1}{2} m\angle ABC = 15^\circ\\ &m\angle QOB = 180^\circ - m\angle OQB - m\angle QBO = 180^\circ - 90^\circ - 15^\circ = 75^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>30°</mcq-option>
<mcq-option>60°</mcq-option>
<mcq-correct>75°</mcq-correct>
<mcq-option>90°</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
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]
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Answer:

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<reasoning>

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

\[

$$\begin{aligned} &OQ \perp AB \implies m\angle OQB = 90^\circ\\ &m\angle QBO = m\angle ABO = \frac{1}{2} m\angle ABC = 15^\circ\\ &m\angle QOB = 180^\circ - m\angle OQB - m\angle QBO = 180^\circ - 90^\circ - 15^\circ = 75^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>30°</mcq-option>
<mcq-option>60°</mcq-option>
<mcq-correct>75°</mcq-correct>
<mcq-option>90°</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Incenter Theorem"
]
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