Question

linear pairs and vertical angles
what is the value of x?
image of intersecting lines with angles (x + 40)° and (3x)° at point r, and four rectangular options with numbers 20, 60, 35, 70 (order as per image)

Explanation:

Step1: Identify Angle Relationship

The angles \((x + 40)^\circ\) and \((3x)^\circ\) are vertical angles? Wait, no, looking at the diagram, actually, \((x + 40)^\circ\) and \((3x)^\circ\) are adjacent and form a linear pair? Wait, no, the lines are intersecting, so vertical angles? Wait, no, let's check again. Wait, the two angles \((x + 40)^\circ\) and \((3x)^\circ\) – wait, maybe they are supplementary? Wait, no, if the lines are intersecting, then adjacent angles on a straight line are supplementary. Wait, actually, the angles \((x + 40)^\circ\) and \((3x)^\circ\) – wait, maybe I made a mistake. Wait, no, looking at the diagram, the two angles \((x + 40)^\circ\) and \((3x)^\circ\) are adjacent and form a linear pair? Wait, no, vertical angles are equal. Wait, maybe the angles \((x + 40)^\circ\) and \((3x)^\circ\) are vertical angles? Wait, no, that can't be. Wait, maybe the lines are such that \((x + 40)^\circ\) and \((3x)^\circ\) are supplementary? Wait, no, let's think again. Wait, the problem is about linear pairs and vertical angles. Wait, maybe the two angles \((x + 40)^\circ\) and \((3x)^\circ\) are vertical angles? No, vertical angles are equal. Wait, maybe I misread. Wait, the diagram: two lines intersecting, so the angles opposite each other are vertical angles. Wait, maybe \((x + 40)^\circ\) and \((3x)^\circ\) are supplementary? Wait, no, if they are adjacent and form a linear pair, then their sum is \(180^\circ\). Wait, let's check the options. The options are 20, 60, 35, 70. Let's assume that \((x + 40) + (3x) = 180\)? Wait, no, maybe they are vertical angles? Wait, no, vertical angles are equal. Wait, maybe the angles \((x + 40)^\circ\) and \((3x)^\circ\) are vertical angles? Then \(x + 40 = 3x\), solving \(40 = 2x\), \(x = 20\). Let's check: \(x + 40 = 60\), \(3x = 60\). Yes, that works. So the angle relationship is vertical angles, so they are equal.

Step2: Set Up Equation

Since the angles are vertical angles, they are equal. So:
\(x + 40 = 3x\)

Step3: Solve for \(x\)

Subtract \(x\) from both sides:
\(40 = 2x\)

Divide both sides by 2:
\(x = 20\)

Answer:

20