Question

in \\( \triangle tuv, m\angle t = 98^{\circ} \\) and \\( m\angle u = 25^{\circ} \\). which list has the sides of \\( \triangle tuv \\) in order from longest to shortest?
answer
vt, tu, uv tu, uv, vt
tu, vt, uv uv, tu, vt
uv, vt, tu vt, uv, tu

Explanation:

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

In a triangle, the sum of angles is \(180^{\circ}\). So \(m\angle V=180^{\circ}-m\angle T - m\angle U\). Substituting \(m\angle T = 98^{\circ}\) and \(m\angle U = 25^{\circ}\), we get \(m\angle V=180^{\circ}-98^{\circ}-25^{\circ}=57^{\circ}\).

Step2: Relate angles and sides

In a triangle, the larger the angle, the longer the side opposite it.

• \(\angle T = 98^{\circ}\), side opposite \(\angle T\) is \(UV\).
• \(\angle V = 57^{\circ}\), side opposite \(\angle V\) is \(TU\).
• \(\angle U = 25^{\circ}\), side opposite \(\angle U\) is \(VT\).

Since \(98^{\circ}>57^{\circ}>25^{\circ}\), the order of sides from longest to shortest (opposite to the angles) is \(UV\) (opposite \(\angle T\)), \(VT\) (opposite \(\angle U\)), \(TU\) (opposite \(\angle V\)).

Answer:

UV, VT, TU