Question

in $\triangle rst$, $m\angle r = 102^{\circ}$ and $m\angle s = 45^{\circ}$. which list has the sides of $\triangle rst$ in order from shortest to longest?\
\bigcirc st,tr,rs\
\bigcirc rs,st,tr\
\bigcirc tr,rs,st\
\bigcirc rs, tr, st\
\bigcirc st,rs,tr\
\bigcirc tr,st,rs

Explanation:

Step1: Find measure of ∠T

In a triangle, the sum of interior angles is \(180^\circ\). So, \(m\angle T = 180^\circ - m\angle R - m\angle S\). Substituting \(m\angle R = 102^\circ\) and \(m\angle S = 45^\circ\), we get \(m\angle T = 180 - 102 - 45 = 33^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it. The angles in \(\triangle RST\) are: \(m\angle T = 33^\circ\), \(m\angle S = 45^\circ\), \(m\angle R = 102^\circ\). So, the order of angles from smallest to largest is \(\angle T < \angle S < \angle R\).
The sides opposite these angles are: \(ST\) (opposite \(\angle R\)), \(RT\) (opposite \(\angle S\)), \(RS\) (opposite \(\angle T\)). Wait, correction: Side opposite \(\angle R\) is \(ST\), opposite \(\angle S\) is \(RT\), opposite \(\angle T\) is \(RS\). So, the order of sides from shortest to longest (since smaller angle opposite shorter side) is \(RS\) (opposite \(33^\circ\)), \(RT\) (opposite \(45^\circ\)), \(ST\) (opposite \(102^\circ\))? Wait no, wait: Let's label the triangle properly. In \(\triangle RST\), vertices are \(R\), \(S\), \(T\). So, side opposite \(\angle R\) is \(ST\), side opposite \(\angle S\) is \(RT\), side opposite \(\angle T\) is \(RS\). So angles: \(\angle T = 33^\circ\), \(\angle S = 45^\circ\), \(\angle R = 102^\circ\). So order of angles: \(\angle T < \angle S < \angle R\). Therefore, order of opposite sides: \(RS\) (opposite \(\angle T\)) < \(RT\) (opposite \(\angle S\)) < \(ST\) (opposite \(\angle R\)). Wait, but let's check the options. Wait, maybe I mixed up the sides. Wait, let's re-express:

Angles: \(\angle T = 33^\circ\), \(\angle S = 45^\circ\), \(\angle R = 102^\circ\). So the smallest angle is \(\angle T\) (33°), then \(\angle S\) (45°), then \(\angle R\) (102°). The side opposite \(\angle T\) is \(RS\) (since \(\angle T\) is at vertex \(T\), so side opposite is \(RS\)), side opposite \(\angle S\) is \(RT\) (at vertex \(S\), opposite is \(RT\)), side opposite \(\angle R\) is \(ST\) (at vertex \(R\), opposite is \(ST\)). So, since smaller angle has shorter opposite side, the order of sides from shortest to longest is \(RS\) (opposite 33°), \(RT\) (opposite 45°), \(ST\) (opposite 102°). Wait, but looking at the options: One of the options is \(RS, TR, ST\) (since \(TR\) is same as \(RT\)). So that's the correct order.

Answer:

RS, TR, ST