Question

what is m∠1?
41°
51°
1
m∠1 = □°

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^\circ\). So the third angle (let's call it \(x\)) is \(180^\circ - 41^\circ - 51^\circ\).
\(x = 180 - 41 - 51 = 88^\circ\)

Step2: Find \(m\angle1\) using supplementary angles

\(\angle1\) and \(x\) are supplementary (they form a linear pair), so \(m\angle1 + x = 180^\circ\).
\(m\angle1 = 180^\circ - 88^\circ = 92^\circ\) Wait, no, wait. Wait, actually, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So \(m\angle1=41^\circ + 51^\circ\)
\(m\angle1 = 41+51 = 92\)? Wait, no, wait, maybe I made a mistake. Wait, the triangle has angles \(41^\circ\), \(51^\circ\), and the third angle. Then the angle adjacent to \(\angle1\) is the third angle. So the third angle is \(180-(41 + 51)=88\). Then \(\angle1\) and that \(88^\circ\) angle are supplementary? No, wait, no. Wait, the diagram: the line is extended, so \(\angle1\) is an exterior angle. The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So the two non - adjacent interior angles are \(41^\circ\) and \(51^\circ\). So \(m\angle1=41 + 51=92^\circ\)? Wait, no, wait, \(41+51 = 92\)? Wait, \(41 + 51=92\)? Wait, \(41+51 = 92\)? Wait, no, \(41+51 = 92\)? Wait, \(41 + 51=92\)? Wait, no, \(41+51 = 92\)? Wait, maybe I messed up. Wait, let's recalculate. The sum of the two non - adjacent interior angles: \(41+51 = 92\). So the exterior angle \(\angle1\) is equal to that sum. So \(m\angle1 = 92^\circ\). Wait, but let's check with the triangle angle sum. The third angle in the triangle is \(180-(41 + 51)=88\). Then the angle adjacent to \(\angle1\) is \(88\), so \(\angle1\) and \(88\) are supplementary? No, that would be \(180 - 88=92\). Yes, so that's correct.

Answer:

\(92\)