Question

given that ( pparallel q ) and ( rparallel s ), find the measure of ( angle15 ).
a. ( 25^{circ} )
b. ( 35^{circ} )
c. ( 65^{circ} )
d. ( 90^{circ} )

Explanation:

Step1: Find the measure of ∠5

Since \(r\parallel s\) and the sum of angles on a straight line is \(180^{\circ}\), and one angle is \(65^{\circ}\) and the right - angle (\(90^{\circ}\)) is also there. But wait, another way: The angle adjacent to \(65^{\circ}\) (let's call it \(x\)) forms a right - angle with another part. Wait, no. Since \(r\parallel s\) and we consider the transversal. The angle adjacent to \(65^{\circ}\) (in the upper part) and the angle we need (related to \(\angle15\)) are alternate interior angles. First, note that the angle adjacent to \(65^{\circ}\) (in the upper part where the right - angle is) is \(25^{\circ}\) (because \(90 - 65=25\)). But actually, using the property of parallel lines \(r\parallel s\) and the transversal. The angle \(\angle5\) and the \(65^{\circ}\) angle: \(\angle5 + 65^{\circ}+90^{\circ}=180^{\circ}\) (sum of angles around a point on one side of a line). Wait, no. Wait, the vertical angle of \(\angle15\) (let's use the property of parallel lines \(r\parallel s\)). The angle \(\angle5\) and \(\angle15\) are alternate interior angles. The angle adjacent to \(65^{\circ}\) (in the upper part) forms a right - triangle - like part. Wait, another approach: Since \(r\parallel s\), and we know that the angle \(\angle5\) and \(\angle15\) are congruent (alternate interior angles). The angle adjacent to \(65^{\circ}\) (in the upper part) is \(25^{\circ}\) (because \(90 - 65 = 25\))? No. Wait, the sum of angles in a right - angled situation (the upper part where \(65^{\circ}\) is). Wait, actually, since \(r\parallel s\), and we consider the transversal. The angle \(\angle5\) (which is \(25^{\circ}\), because \(90^{\circ}-65^{\circ}=25^{\circ}\)) and \(\angle15\) are alternate interior angles.

Step2: Calculate \(\angle15\)

Since \(r\parallel s\) (given) and considering the transversal, \(\angle15=\angle5\). The angle \(\angle5\) is \(25^{\circ}\) (because \(90^{\circ}-65^{\circ} = 25^{\circ}\), as the sum of the two non - right angles in a right - angled part (formed by the intersection of lines) is \(90^{\circ}\)).

Answer:

A. \(25^{\circ}\)