List the side lengths of Delta XYZ in order from smallest to largest, given that mangle Z=38^circ and mangle X=79^circ square lt square lt square

To determine the side lengths of \(\Delta XYZ\) in order from smallest to largest, we first need to find the measure of the third angle, \(m\angle Y\).<br /><br />In any triangle, the sum of the interior angles is \(180^\circ\). Therefore, we can calculate \(m\angle Y\) as follows:<br /><br />\[<br />m\angle Y = 180^\circ - m\angle X - m\angle Z = 180^\circ - 79^\circ - 38^\circ = 63^\circ<br />\]<br /><br />Now, we have the measures of all three angles:<br />- \(m\angle X = 79^\circ\)<br />- \(m\angle Y = 63^\circ\)<br />- \(m\angle Z = 38^\circ\)<br /><br />The side opposite the largest angle will be the longest, and the side opposite the smallest angle will be the shortest. Thus, we compare the angles:<br /><br />1. \(m\angle X = 79^\circ\) (largest angle)<br />2. \(m\angle Y = 63^\circ\)<br />3. \(m\angle Z = 38^\circ\) (smallest angle)<br /><br />Therefore, the sides opposite these angles, in order from smallest to largest, are:<br /><br />- Side opposite \(\angle Z\) (smallest angle)<br />- Side opposite \(\angle Y\)<br />- Side opposite \(\angle X\) (largest angle)<br /><br />Thus, the side lengths of \(\Delta XYZ\) in order from smallest to largest are:<br /><br />\[<br />YZ < XZ < XY<br />\]