Question

in \\( \triangle stu, m \angle s = 51 ^ { \circ } \\) and \\( m \angle t = 28 ^ { \circ } \\). which statement about the sides of \\( \triangle stu \\) must be true?
answer
\\( \bigcirc st > tu > us \\)
\\( \bigcirc tu > st > us \\)
\\( \bigcirc us > st > tu \\)
\\( \bigcirc st > us > tu \\)
\\( \bigcirc us > tu > st \\)
\\( \bigcirc tu > us > st \\)

Explanation:

Step1: Find the measure of angle \( U \)

The sum of angles in a triangle is \( 180^{\circ} \). So, \( m\angle U=180^{\circ}-m\angle S - m\angle T \).
Substitute \( m\angle S = 51^{\circ} \) and \( m\angle T = 28^{\circ} \):
\( m\angle U=180^{\circ}-51^{\circ}-28^{\circ}=101^{\circ} \).

Step2: Relate angles and sides

In a triangle, the larger the angle, the longer the side opposite it.
Since \( m\angle U(101^{\circ})>m\angle S(51^{\circ})>m\angle T(28^{\circ}) \), the sides opposite these angles follow the same order.
Side opposite \( \angle U \) is \( ST \), side opposite \( \angle S \) is \( TU \), and side opposite \( \angle T \) is \( US \).
So, \( ST>TU>US \).

Answer:

ST > TU > US