Question

find the measures of the labeled angles.
(x + 105)° 4x°
(x + 105)° = 140 °
(type a whole number.)
4x° = □°
(type a whole number.)

Explanation:

Step1: Find x from given angle

We know that \((x + 105)^\circ=140^\circ\). Subtract 105 from both sides:
\(x=140 - 105\)
\(x = 35\)

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

Substitute \(x = 35\) into \(4x\):
\(4x=4\times35\)
\(4x = 140\)? Wait, no, wait. Wait, actually, the two angles are supplementary? Wait, no, wait, when two lines intersect, adjacent angles are supplementary? Wait, no, wait, the given angle \((x + 105)^\circ = 140^\circ\), so first find x. Wait, \(x+105 = 140\), so \(x=35\). Then \(4x=4\times35 = 140\)? Wait, but that would mean both angles are 140, but that can't be. Wait, maybe I made a mistake. Wait, no, wait, maybe the two angles are supplementary? Wait, no, if two lines intersect, adjacent angles are supplementary (sum to 180). Wait, but the problem says \((x + 105)^\circ=140^\circ\), so let's check. If \((x + 105)=140\), then \(x = 35\), then \(4x=140\). But then \(140 + 140=280\), which is more than 180. Wait, maybe the two angles are vertical angles? Wait, no, vertical angles are equal. Wait, maybe the diagram is of two intersecting lines, and the two angles given are adjacent and supplementary. Wait, but the problem states \((x + 105)^\circ = 140^\circ\), so we can use that to find x, then find \(4x\). Wait, let's do that.

Wait, the problem says \((x + 105)^\circ = 140^\circ\), so solve for x:

\(x+105 = 140\)

Subtract 105 from both sides:

\(x=140 - 105=35\)

Then \(4x=4\times35 = 140\)? But that would mean both angles are 140, which would imply they are vertical angles (since vertical angles are equal). Wait, maybe the diagram shows vertical angles. So if \((x + 105)^\circ\) and \(4x^\circ\) are vertical angles, then they should be equal. So \((x + 105)=4x\), but the problem already gives \((x + 105)^\circ = 140^\circ\), so \(4x\) should also be 140? Wait, that seems contradictory, but maybe the problem has a typo, or maybe I misinterpret the diagram. Wait, the problem says "Find the measures of the labeled angles" and gives \((x + 105)^\circ = 140^\circ\) (already filled) and asks for \(4x^\circ\). So using \(x = 35\) (from \(x + 105 = 140\)), then \(4x=4\times35 = 140\). Wait, but that would mean both angles are 140, which are vertical angles (since vertical angles are equal). So maybe that's the case.

Answer:

\(4x^\circ=\boxed{140}^\circ\)