Question

use the diagram below to answer the questions. find ( mangle rpt ).

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. The angle of \(140^{\circ}\) and the angle adjacent to \((70m)^{\circ}\) form a linear - pair. So the angle adjacent to \((70m)^{\circ}\) is \(180 - 140=40^{\circ}\). Also, we know that the sum of angles around a point is \(360^{\circ}\). But another way is to use the property of vertical angles. The \(140^{\circ}\) angle and the angle \((70m)^{\circ}\) are related as follows: Since \(140^{\circ}\) and \((70m)^{\circ}\) are vertical angles (no, wait, correction: they are adjacent to vertical - angle pairs. Actually, using the linear - pair property. The angle of \(140^{\circ}\) and the angle \(\angle RPT\) are vertical angles. Wait, no. Wait, the \(140^{\circ}\) angle and the angle adjacent to \((70m)^{\circ}\) form a linear pair. But actually, we know that vertical angles are equal. The \(140^{\circ}\) angle and the angle opposite to \((70m)^{\circ}\) (not in the diagram's given labels, but using the property that \(140 = 70m\) (because vertical angles are equal)).
Set up the equation \(70m=140\).

Step2: Solve for \(m\)

Divide both sides of the equation \(70m = 140\) by \(70\). So \(m=\frac{140}{70}=2\). Then \(\angle RPT=(70m)^{\circ}\). Substitute \(m = 2\) into \((70m)^{\circ}\), we get \(\angle RPT=140^{\circ}\). Wait, no, wait, another approach: The \(140^{\circ}\) angle and \(\angle RPT\) are vertical angles. By the vertical - angles theorem (vertical angles are congruent), \(\angle RPT = 140^{\circ}\).

Answer:

\(140^{\circ}\)