Question

find the measures of the labeled angles.
(x + 102)° 4x°

(x + 102)° = □°
(type a whole number.)

Explanation:

Step1: Identify angle relationship

The two angles are vertical angles? No, wait, they are adjacent and form a linear pair? Wait, no, when two lines intersect, adjacent angles are supplementary? Wait, no, the two angles given are \((x + 102)^\circ\) and \(4x^\circ\). Wait, actually, when two lines intersect, the adjacent angles are supplementary (sum to \(180^\circ\))? Wait, no, maybe they are vertical angles? Wait, no, looking at the diagram, the two angles are adjacent and form a linear pair? Wait, no, maybe they are supplementary? Wait, no, actually, if two lines intersect, the angles that are adjacent (form a linear pair) sum to \(180^\circ\). Wait, but in this case, maybe the two angles are supplementary? Wait, no, let's think again. Wait, the two angles \((x + 102)^\circ\) and \(4x^\circ\) are adjacent and form a linear pair, so their sum is \(180^\circ\)? Wait, no, maybe they are vertical angles? Wait, no, vertical angles are equal. Wait, maybe I made a mistake. Wait, the diagram shows two intersecting lines, so the two angles labeled \((x + 102)^\circ\) and \(4x^\circ\) are adjacent and form a linear pair? Wait, no, actually, when two lines intersect, the adjacent angles are supplementary (sum to \(180^\circ\))? Wait, no, vertical angles are equal. Wait, maybe the two angles are supplementary? Wait, let's check the problem again.

Wait, the problem is to find the measure of \((x + 102)^\circ\). Let's assume that the two angles are supplementary (since they form a linear pair). So:

\((x + 102) + 4x = 180\)

Step2: Solve for x

Combine like terms:

\(5x + 102 = 180\)

Subtract 102 from both sides:

\(5x = 180 - 102\)

\(5x = 78\)? Wait, no, 180 - 102 is 78? Wait, 180 - 100 is 80, minus 2 is 78. Then \(x = 78 / 5 = 15.6\)? But the problem says to type a whole number. So maybe my assumption is wrong. Wait, maybe the two angles are vertical angles? So \((x + 102) = 4x\)

Let's try that:

\(x + 102 = 4x\)

Subtract x from both sides:

\(102 = 3x\)

Divide both sides by 3:

\(x = 34\)

Then \((x + 102) = 34 + 102 = 136\), and \(4x = 4*34 = 136\). So that works. So the two angles are vertical angles, so they are equal. So that's the correct approach.

So step1: Set the angles equal (vertical angles are equal)

\(x + 102 = 4x\)

Step2: Solve for x

\(102 = 4x - x\)

\(102 = 3x\)

\(x = 102 / 3 = 34\)

Step3: Find \((x + 102)^\circ\)

Substitute x = 34:

\(34 + 102 = 136\)

Answer:

136