Question

angle 1
70
angle 2
choose
65
40
45
70
140
115
135
angle 3
angle 4
angle 5
choose
angle 6
choose
angle 7
choose

Explanation:

Step1: Find the measure of Angle 2

Since Angle 2 and the \(115^{\circ}\) angle are supplementary (they form a linear - pair), we use the formula \(\angle2 = 180^{\circ}-\text{given angle}\).
\(\angle2=180 - 115=65^{\circ}\)

Step2: Find the measure of Angle 3

Angle 2 and Angle 3 are alternate - interior angles. When two parallel lines are cut by a transversal, alternate - interior angles are equal. So \(\angle3=\angle2 = 65^{\circ}\)

Step3: Find the measure of Angle 4

Angle 4 and the \(115^{\circ}\) angle are vertical angles. Vertical angles are equal. So \(\angle4 = 115^{\circ}\)

Step4: Find the measure of Angle 6

First, find the sum of the angles in the triangle - like part formed by Angle 1, Angle 2, and Angle 7. We know that the sum of angles in a triangle is \(180^{\circ}\). But we can also use the property of parallel lines. Since \(q\parallel m\) (assuming \(q\) and \(m\) are parallel), and using the transversal, we know that \(\angle6\) and \(\angle3\) are supplementary (consecutive - interior angles). \(\angle6=180 - 70=110^{\circ}\) (wait, no, let's re - do.

We know that \(\angle1 = 70^{\circ}\), \(\angle2=65^{\circ}\). In the triangle - like figure (if we consider the intersection of the two transversals), we can also use the property of parallel lines. Another way: \(\angle7\) and \(\angle1\) are related. \(\angle7\) and \(\angle1\) are alternate - interior angles? No. Wait, \(\angle1\) and the \(110^{\circ}\) angle are supplementary (linear - pair), \(\angle1=70^{\circ}\).

Since \(q\parallel m\) (assuming \(q\) and \(m\) are parallel), \(\angle2\) and \(\angle4\) are related as \(\angle4 = 115^{\circ}\) (vertical angles with the \(115^{\circ}\) angle). \(\angle7\) and \(\angle1\) form a triangle - like part. But using the property of parallel lines: \(\angle7\) and \(\angle1\) (if we consider the transversal). Wait, \(\angle7\) and \(\angle1\) are not in a direct parallel - line relationship.

Another approach: \(\angle2 = 65^{\circ}\), \(\angle1=70^{\circ}\). In the triangle (formed by the intersection of the two non - parallel lines and line \(q\)), the third angle (let's call it \(x\)) has \(x=180-(70 + 65)=45^{\circ}\). Then \(\angle7=x = 45^{\circ}\) (vertical angles). And \(\angle6\) and \(\angle7\) are supplementary (linear - pair), so \(\angle6=180 - 45=135^{\circ}\)

Step5: Find the measure of Angle 5

Since \(q\parallel m\) and using the transversal, \(\angle5\) and \(\angle6\) are alternate - interior angles. So \(\angle5=\angle6 = 135^{\circ}\)

Answer:

Angle 2: \(65\), Angle 3: \(65\), Angle 4: \(115\), Angle 5: \(135\), Angle 6: \(135\), Angle 7: \(45\)