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in \\( \triangle \mathrm { abc }, \mathrm { bc } = 5, \mathrm { ca } = 9 \\), and \\( \mathrm { ab } = 8 \\). which
statement about the angles of \\( \triangle \mathrm { abc } \\) must be true?
answer
\\( \mathrm { m } \angle c < \mathrm { m } \angle b < \mathrm { m } \angle a \\)
\\( \mathrm { m } \angle b < \mathrm { m } \angle a < \mathrm { m } \angle c \\)
\\( \mathrm { m } \angle a < \mathrm { m } \angle c < \mathrm { m } \angle b \\)
\\( \mathrm { m } \angle b < \mathrm { m } \angle c < \mathrm { m } \angle a \\)
\\( \mathrm { m } \angle a < \mathrm { m } \angle b < \mathrm { m } \angle c \\)
\\( \mathrm { m } \angle c < \mathrm { m } \angle a < \mathrm { m } \angle b \\)
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answer

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. That is, if in \(\triangle ABC\), side \(a\) is opposite \(\angle A\), side \(b\) is opposite \(\angle B\), and side \(c\) is opposite \(\angle C\), then \(a > b>c\) implies \(m\angle A>m\angle B > m\angle C\) (and vice - versa).

First, we need to identify the sides and their opposite angles:

• In \(\triangle ABC\), side \(BC\) is opposite \(\angle A\), so \(BC = a=5\), opposite \(\angle A\).
• Side \(AC\) is opposite \(\angle B\), so \(AC = b = 9\), opposite \(\angle B\).
• Side \(AB\) is opposite \(\angle C\), so \(AB=c = 8\), opposite \(\angle C\).

Step2: Compare the lengths of the sides

We have the side lengths: \(BC = 5\), \(AB = 8\), \(AC=9\). So, \(BC

Step3: Relate side lengths to angle measures

Since the side opposite \(\angle A\) is \(BC = 5\), the side opposite \(\angle C\) is \(AB = 8\), and the side opposite \(\angle B\) is \(AC=9\), and using the angle - side relationship (larger side opposite larger angle), we get:
\(m\angle A

Answer:

\(m\angle A < m\angle C < m\angle B\) (the option: \(m\angle A < m\angle C < m\angle B\))