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 largest side of this triangle?

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

Explanation:

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<reasoning>

Identify the given angle measurements

We are given the interior angles of \(\triangle ABC\):
\[

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

\]

Determine the largest angle

Comparing the angle measures:
\[
75^\circ > 60^\circ > 45^\circ
\]
Thus, \(\angle C\) is the largest angle.

Apply the angle-side relationship

In any triangle, the longest side is always opposite the largest angle.

• The side opposite to \(\angle A\) is \(\overline{BC}\).
• The side opposite to \(\angle B\) is \(\overline{AC}\).
• The side opposite to \(\angle C\) is \(\overline{AB}\).

Since \(\angle C\) is the largest angle, the opposite side \(\overline{AB}\) must be the longest side.
</reasoning>

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

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Answer:

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<reasoning>

Identify the given angle measurements

We are given the interior angles of \(\triangle ABC\):
\[

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

\]

Determine the largest angle

Comparing the angle measures:
\[
75^\circ > 60^\circ > 45^\circ
\]
Thus, \(\angle C\) is the largest angle.

Apply the angle-side relationship

In any triangle, the longest side is always opposite the largest angle.

• The side opposite to \(\angle A\) is \(\overline{BC}\).
• The side opposite to \(\angle B\) is \(\overline{AC}\).
• The side opposite to \(\angle C\) is \(\overline{AB}\).

Since \(\angle C\) is the largest angle, the opposite side \(\overline{AB}\) must be the longest side.
</reasoning>

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

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