Question

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list the angle measures of δrst in order from smallest to largest. assume that p is a positive number.

m∠□ < m∠□ < m∠□

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. First, we need to compare the lengths of the sides to determine the order of the angles. The sides are \(29p\), \(67p\), and we can find the third side's length or use the angle sum property. Wait, actually, we know one angle is \(93^\circ\), and the other two angles are related to the sides \(29p\) (opposite \(\angle T\)) and \(67p\) (opposite \(\angle S\)), and the side opposite \(93^\circ\) (which is \(ST\))? Wait, no, let's label the triangle: vertices \(R\), \(S\), \(T\). Side \(RS = 29p\) (opposite \(\angle T\)), side \(RT = 67p\) (opposite \(\angle S\)), and side \(ST\) is opposite \(\angle R = 93^\circ\).

First, compare the lengths of \(RS = 29p\) and \(RT = 67p\). Since \(p>0\), \(29p < 67p\). So the side opposite \(\angle T\) (which is \(RS = 29p\)) is shorter than the side opposite \(\angle S\) (which is \(RT = 67p\)). Therefore, \(\angle T < \angle S\) (because smaller side opposite smaller angle). Now, we know one angle is \(93^\circ\) (\(\angle R\)). Let's check the sum of the other two angles: the sum of angles in a triangle is \(180^\circ\), so \(\angle S + \angle T + 93^\circ = 180^\circ\), so \(\angle S + \angle T = 87^\circ\). Wait, that can't be, because if \(\angle R = 93^\circ\), which is obtuse, then the other two angles must be acute (since a triangle can have only one obtuse angle). Wait, maybe I mixed up the sides. Wait, let's re-express:

Wait, the side adjacent to \(\angle R\) (93°) is \(RS = 29p\) and \(RT = 67p\), so the side opposite \(\angle R\) is \(ST\). The side opposite \(\angle S\) is \(RT = 67p\), and the side opposite \(\angle T\) is \(RS = 29p\). So since \(29p < 67p\), then \(\angle T < \angle S\) (because side opposite \(\angle T\) is shorter than side opposite \(\angle S\)). Now, \(\angle R = 93^\circ\), which is the largest angle (since it's obtuse, and the other two are acute, summing to \(87^\circ\)). So the order of angles from smallest to largest: \(\angle T\) (opposite \(29p\)), \(\angle S\) (opposite \(67p\)), and then \(\angle R\) (93°). Wait, but let's confirm with the angle sum. Wait, no, wait: if \(\angle R = 93^\circ\), then \(\angle S + \angle T = 180 - 93 = 87^\circ\). So both \(\angle S\) and \(\angle T\) are less than \(93^\circ\), and since \(29p < 67p\), \(\angle T\) (opposite \(29p\)) is smaller than \(\angle S\) (opposite \(67p\)). Therefore, the angles in order from smallest to largest: \(\angle T\), \(\angle S\), \(\angle R\).

Wait, let's check the side lengths again. \(RS = 29p\) (opposite \(\angle T\)), \(RT = 67p\) (opposite \(\angle S\)), and \(ST\) (opposite \(\angle R = 93^\circ\)). Since \(29p < 67p\), then \(\angle T < \angle S\). And since \(\angle R = 93^\circ\) is obtuse, it's the largest angle. So the order is \(m\angle T < m\angle S < m\angle R\).

Step2: Confirm the Angles

So the angles are \(\angle T\) (opposite \(29p\)), \(\angle S\) (opposite \(67p\)), and \(\angle R = 93^\circ\). Since \(29p < 67p\), \(\angle T < \angle S\), and \(\angle R = 93^\circ\) is larger than both (since \(\angle S + \angle T = 87^\circ\), so both are less than 93°). Therefore, the order from smallest to largest is \(m\angle T < m\angle S < m\angle R\).

Answer:

\(m\angle T < m\angle S < m\angle R\)