Question

problem 11: spiral review. use what you know about the interior - angle measures of triangles to complete this problem. triangle pqr is an obtuse triangle and the measure of angle q is 72°. what are possible measures of the other two angles? angle p angle r

Explanation:

Step1: Recall triangle - angle sum property

The sum of the interior angles of a triangle is 180°. Let \(\angle Q = 72^{\circ}\), and \(\angle P\) and \(\angle R\) be the other two angles. So \(\angle P+\angle R+\angle Q=180^{\circ}\), which means \(\angle P + \angle R=180^{\circ}-\angle Q = 180 - 72=108^{\circ}\).

Step2: Consider the condition for an obtuse - triangle

Since \(\triangle PQR\) is an obtuse - triangle, one of the angles \(\angle P\) or \(\angle R\) must be greater than 90°. Let \(\angle P>90^{\circ}\), then \(\angle R = 108^{\circ}-\angle P\). Also, since \(\angle R>0^{\circ}\), we have \(0^{\circ}<108^{\circ}-\angle P\), so \(\angle P < 108^{\circ}\). And since \(\angle R=108^{\circ}-\angle P\), when \(\angle P>90^{\circ}\), \(\angle R < 18^{\circ}\).
If \(\angle R>90^{\circ}\), then \(\angle P=108^{\circ}-\angle R\), and \(0^{\circ}<108^{\circ}-\angle R\), so \(\angle R < 108^{\circ}\), and \(\angle P<18^{\circ}\).

So the possible measures:
If \(\angle P>90^{\circ}\), then \(90^{\circ}<\angle P<108^{\circ}\) and \(0^{\circ}<\angle R < 18^{\circ}\).
If \(\angle R>90^{\circ}\), then \(90^{\circ}<\angle R<108^{\circ}\) and \(0^{\circ}<\angle P < 18^{\circ}\)

Answer:

If \(\angle P\) is the obtuse angle: \(90^{\circ}<\angle P<108^{\circ}\), \(0^{\circ}<\angle R < 18^{\circ}\); if \(\angle R\) is the obtuse angle: \(90^{\circ}<\angle R<108^{\circ}\), \(0^{\circ}<\angle P < 18^{\circ}\)