Question

1. find the value of x

(diagram: two vertical parallel lines cut by a transversal, with angles 80° and ((x + 15)^circ))
find the measure of ( x )
a) ( x = 165^circ ) b) ( x = 65^circ )
c) ( x = 85^circ ) d) ( x = 75^circ )

Explanation:

Step1: Identify angle relationship

The two vertical lines are parallel, and the transversal creates same - side interior angles? Wait, no, actually, the \(80^{\circ}\) and \((x + 15)^{\circ}\) are supplementary? Wait, no, looking at the diagram (parallel lines cut by a transversal), the \(80^{\circ}\) and \((x + 15)^{\circ}\) should be supplementary? Wait, no, actually, if the lines are parallel, the consecutive interior angles are supplementary. Wait, no, let's think again. Wait, the two angles \(80^{\circ}\) and \((x + 15)^{\circ}\) are adjacent and form a linear pair? No, wait, the two vertical lines are parallel, so the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are supplementary? Wait, no, actually, if we consider that the sum of the two angles \(80^{\circ}\) and \((x + 15)^{\circ}\) is \(180^{\circ}\)? Wait, no, that can't be. Wait, maybe they are alternate interior angles? No, wait, the diagram shows two parallel vertical lines cut by a transversal. So the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are supplementary? Wait, no, let's check the sum. Wait, if we assume that \(80+(x + 15)=180\)? No, that would give \(x=85\), but that's option c. Wait, no, maybe I made a mistake. Wait, no, wait, the two angles are actually supplementary? Wait, no, let's do the calculation.

Wait, the correct relationship: since the two lines are parallel, the consecutive interior angles are supplementary. Wait, the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are consecutive interior angles? Wait, no, maybe they are same - side interior angles. So \(80+(x + 15)=180\)? Wait, solving for \(x\):

Step2: Solve the equation

\(80+(x + 15)=180\)

First, simplify the left - hand side: \(x+95 = 180\)

Then, subtract 95 from both sides: \(x=180 - 95=85\)? Wait, no, that's option c. But wait, maybe I misread the diagram. Wait, maybe the two angles are complementary? No, that doesn't make sense. Wait, no, maybe the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are supplementary? Wait, no, let's check the options. Option b is \(x = 65\), option d is \(x = 75\), option a is \(165\), option c is \(85\).

Wait, maybe the correct relationship is that the two angles are supplementary? Wait, no, let's re - examine. If the two lines are parallel, and the transversal cuts them, then the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are supplementary? Wait, no, \(80+(x + 15)=180\) gives \(x = 85\), which is option c. But wait, maybe the diagram is such that the two angles are supplementary. Wait, let's do the math again.

\(80+(x + 15)=180\)

\(x+95 = 180\)

\(x=180 - 95\)

\(x = 85\)

Wait, but let's check the options. Option c is \(x = 85^{\circ}\).

Wait, maybe I made a mistake in the angle relationship. Wait, maybe the two angles are alternate interior angles? No, alternate interior angles are equal. If they were equal, \(80=x + 15\), then \(x=65\), which is option b. Oh! Wait, maybe the lines are parallel, so the alternate interior angles are equal. So \(80=x + 15\)? Wait, that would be the case if the angles are alternate interior angles. Let's check:

If the two vertical lines are parallel, and the transversal cuts them, then the angle \(80^{\circ}\) and \((x + 15)^{\circ}\) are alternate interior angles, so they are equal. So:

Step1: Set up the equation (alternate interior angles)

Since the lines are parallel, alternate interior angles are equal. So \(80=x + 15\)

Step2: Solve for x

Subtract 15 from both sides: \(x=80 - 15=65\)

Ah! That makes sense. So the correct equation is \(80=x + 15\) because the angles are alternate interior angles (since the two vertical lines are parallel, the transversal creates alternate interior angles that are equal).

So \(x=80 - 15 = 65\)

Answer:

b) \(x = 65^{\circ}\)