Question

find the measures of the labeled angles.
(x + 102)° 4x°
(x + 102)° = 136 °
(type a whole number.)
4x° = □°
(type a whole number.)

Explanation:

Step1: Find x from given angle

We know that \((x + 102)^\circ=136^\circ\). Solve for \(x\):
\(x=136 - 102\)
\(x = 34\)

Step2: Calculate \(4x^\circ\)

Substitute \(x = 34\) into \(4x\):
\(4x=4\times34\)
\(4x = 136\)? Wait, no, wait. Wait, actually, the two angles \((x + 102)^\circ\) and \(4x^\circ\) are supplementary? Wait, no, they are vertical angles? Wait, no, when two lines intersect, adjacent angles are supplementary. Wait, the given \((x + 102)^\circ\) and \(4x^\circ\) are adjacent angles forming a linear pair, so they should be supplementary. Wait, but the first angle is given as \(136^\circ\). Wait, let's check again.

Wait, the problem says \((x + 102)^\circ = 136^\circ\), so we can find \(x\) first.

From \((x + 102)=136\), subtract 102 from both sides: \(x=136 - 102=34\).

Now, substitute \(x = 34\) into \(4x\): \(4x=4\times34 = 136\)? Wait, that can't be, because if two adjacent angles are supplementary, their sum should be \(180^\circ\). Wait, maybe I made a mistake. Wait, no, maybe the two angles are vertical angles? Wait, no, the diagram shows two intersecting lines, so adjacent angles are supplementary, vertical angles are equal. Wait, the problem says \((x + 102)^\circ = 136^\circ\), so let's check: if \((x + 102)=136\), then \(x = 34\), then \(4x=136\), but \(136+136 = 272\), which is more than \(180\). Wait, that's a problem. Wait, maybe the two angles are supplementary. So \((x + 102)+4x=180\). Let's solve that: \(5x+102 = 180\), \(5x=78\), \(x = 15.6\), but the first angle is given as \(136^\circ\), so \((x + 102)=136\) implies \(x = 34\), then \(4x=136\), so maybe the two angles are vertical angles, so they are equal. So if \((x + 102)=136\), then \(4x\) should also be \(136\)? Wait, but \(4x\) when \(x = 34\) is \(136\), so that's correct. So the answer is \(136\)? Wait, but that would mean the two angles are equal, so they are vertical angles. So the calculation is:

Given \((x + 102)^\circ=136^\circ\), so \(x=136 - 102 = 34\). Then \(4x^\circ=4\times34^\circ = 136^\circ\).

Answer:

\(136\)