Question

\\(\delta abc\\) has the following angle measurements:

\\(\angle a = 45^{\circ}\\)
\\(\angle b = 60^{\circ}\\)
\\(\angle c = 75^{\circ}\\)

which of the following is the smallest side of this triangle?

• \\(\overline{ac}\\)
• \\(\overline{ab}\\)
• \\(\overline{ba}\\)
• \\(\overline{bc}\\)

Explanation:

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<reasoning>

Identify the given angle measurements

\[

$$\begin{aligned} \angle A &= 45^\circ \\ \angle B &= 60^\circ \\ \angle C &= 75^\circ \end{aligned}$$

\]

Compare the angle sizes to find the smallest angle

\[
45^\circ < 60^\circ < 75^\circ \implies \angle A < \angle B < \angle C
\]
The smallest angle is \(\angle A\).

Determine the side opposite to the smallest angle

In \(\triangle ABC\), the side opposite to \(\angle A\) is \(\overline{BC}\).
By the angle-side relationship in a triangle, the shortest side is opposite the smallest angle.
Therefore, the smallest side is \(\overline{BC}\).
</reasoning>

<answer>
<mcq-option>(A) \(\overline{AC}\)</mcq-option>
<mcq-option>(B) \(\overline{AB}\)</mcq-option>
<mcq-option>(C) \(\overline{BA}\)</mcq-option>
<mcq-correct>(D) \(\overline{BC}\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Angle-Side Relationship"
]
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Answer:

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<reasoning>

Identify the given angle measurements

\[

$$\begin{aligned} \angle A &= 45^\circ \\ \angle B &= 60^\circ \\ \angle C &= 75^\circ \end{aligned}$$

\]

Compare the angle sizes to find the smallest angle

\[
45^\circ < 60^\circ < 75^\circ \implies \angle A < \angle B < \angle C
\]
The smallest angle is \(\angle A\).

Determine the side opposite to the smallest angle

In \(\triangle ABC\), the side opposite to \(\angle A\) is \(\overline{BC}\).
By the angle-side relationship in a triangle, the shortest side is opposite the smallest angle.
Therefore, the smallest side is \(\overline{BC}\).
</reasoning>

<answer>
<mcq-option>(A) \(\overline{AC}\)</mcq-option>
<mcq-option>(B) \(\overline{AB}\)</mcq-option>
<mcq-option>(C) \(\overline{BA}\)</mcq-option>
<mcq-correct>(D) \(\overline{BC}\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Angle-Side Relationship"
]
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