Question

in this exercise, lines m and n are parallel. find the measure of each numbered angle. the figure is not to scale. m∠1 = 80° m∠2 = 45° m∠3 = 55° m∠4 = \boxed{}°

Explanation:

Step1: Identify angle relationship

The angle of \(80^\circ\) and \(\angle 4\) are adjacent and form a linear pair? Wait, no, looking at the diagram, the \(80^\circ\) angle and \(\angle 4\) – actually, wait, the straight line (line \(m\)) and the transversal: the \(80^\circ\) angle and \(\angle 4\) are adjacent? Wait, no, let's see: the angle marked \(80^\circ\) and \(\angle 4\) – wait, actually, the angle of \(80^\circ\) and \(\angle 4\) are complementary? No, wait, the straight line (line \(m\)): the sum of angles on a straight line is \(180^\circ\)? Wait, no, the \(80^\circ\) angle and \(\angle 4\) – wait, maybe vertical angles or adjacent. Wait, no, the angle of \(80^\circ\) and \(\angle 4\): wait, the diagram shows that the angle adjacent to \(80^\circ\) and \(\angle 4\) – wait, actually, the \(80^\circ\) angle and \(\angle 4\) are such that \(80^\circ + \angle 4 + \) another angle? Wait, no, looking at the intersection: the angle of \(80^\circ\) and \(\angle 4\) – wait, maybe the \(80^\circ\) angle and \(\angle 4\) are adjacent and form a linear pair? No, wait, the straight line is line \(m\), so the sum of angles on a straight line is \(180^\circ\). Wait, the angle of \(80^\circ\) and \(\angle 4\) – wait, no, the angle marked \(80^\circ\) and \(\angle 4\) are vertical? No, wait, let's re-examine. The angle of \(80^\circ\) and \(\angle 4\): actually, the \(80^\circ\) angle and \(\angle 4\) are adjacent, and the angle opposite to \(80^\circ\) is \(\angle 1\) (which is \(80^\circ\) as given). Then, the angle \(\angle 4\): wait, the sum of \(80^\circ\) and \(\angle 4\) and the right angle? No, wait, maybe the \(80^\circ\) angle and \(\angle 4\) are complementary? Wait, no, the diagram: the angle of \(80^\circ\) and \(\angle 4\) – wait, actually, the angle of \(80^\circ\) and \(\angle 4\) are such that \(80^\circ + \angle 4 = 180^\circ - \) something? Wait, no, let's think again. The angle of \(80^\circ\) and \(\angle 4\) are adjacent, and the line is straight, so \(80^\circ + \angle 4 + \) another angle? Wait, no, the angle of \(80^\circ\) and \(\angle 4\) – wait, maybe the \(80^\circ\) angle and \(\angle 4\) are vertical? No, the given \(m\angle 1 = 80^\circ\), which is vertical to the \(80^\circ\) angle. Then, the angle \(\angle 4\): let's see, the angle of \(125^\circ\) and its adjacent angle on line \(n\) is \(180 - 125 = 55^\circ\). Then, since lines \(m\) and \(n\) are parallel, the corresponding angle to that \(55^\circ\) on line \(m\) would be equal. Wait, but maybe \(\angle 4\) is related to the \(80^\circ\) angle. Wait, the \(80^\circ\) angle and \(\angle 4\) – wait, the sum of \(80^\circ\) and \(\angle 4\) and the angle that's equal to \(55^\circ\) (from the \(125^\circ\) angle) should be \(180^\circ\)? Wait, no, let's do step by step.

First, the angle adjacent to \(125^\circ\) on line \(n\) is \(180^\circ - 125^\circ = 55^\circ\). Let's call that angle \(\angle 7\)'s adjacent angle? Wait, \(\angle 7\) and \(125^\circ\) are supplementary, so \(m\angle 7 = 180 - 125 = 55^\circ\). Now, since lines \(m\) and \(n\) are parallel, the corresponding angle to \(\angle 7\) on line \(m\) would be equal. Wait, the angle at line \(m\) and the transversal: the angle that's equal to \(\angle 7\) (55°) would be \(\angle 4\) or \(\angle 5\)? Wait, no, the \(80^\circ\) angle, \(\angle 4\), and the angle equal to \(55^\circ\) (let's say \(\angle 5\)) should add up to \(180^\circ\) because they are on a straight line (line \(m\)). So \(80^\circ + \angle 4 + 55^\circ = 180^\circ\)? Wait, no, that can't be. Wait, maybe the \(80^\circ\) angle and \(\angle 4\) are complementary? Wait, no, let's look at the intersection. The angle of \(80^\circ\) and \(\angle 4\) – wait, the given \(m\angle 1 = 80^\circ\), which is vertical to the \(80^\circ\) angle. Then, the angle \(\angle 4\): let's see, the angle between the two transversals. Wait, maybe the \(80^\circ\) angle and \(\angle 4\) are adjacent, and the angle opposite to the \(55^\circ\) (from the \(125^\circ\) angle) is \(\angle 3\), which is \(55^\circ\) (as given \(m\angle 3 = 55^\circ\)). Then, the sum of \(\angle 3\), \(\angle 4\), and the \(80^\circ\) angle? No, that's not right. Wait, maybe the \(80^\circ\) angle and \(\angle 4\) are such that \(80^\circ + \angle 4 = 180^\circ - 55^\circ\)? No, this is getting confusing. Wait, let's use the fact that on a straight line, the sum of angles is \(180^\circ\). The angle of \(80^\circ\), \(\angle 4\), and the angle equal to \(55^\circ\) (since \(\angle 3 = 55^\circ\), which is vertical to some angle) – wait, \(\angle 3 = 55^\circ\), \(\angle 1 = 80^\circ\), and \(\angle 2\), \(\angle 3\), \(\angle 1\) are angles around a point? No, they are on a straight line? Wait, no, the horizontal line is line \(m\), so the angles on line \(m\) at the intersection: the \(80^\circ\) angle, \(\angle 4\), and the angle adjacent to \(\angle 3\) – wait, maybe \(\angle 4\) is \(180^\circ - 80^\circ - 55^\circ\)? No, that would be \(45^\circ\), but the given \(m\angle 2 = 45^\circ\)? Wait, no, the user's initial input shows \(m\angle 1 = 80^\circ\), \(m\angle 2 = 45^\circ\), \(m\angle 3 = 55^\circ\), and \(m\angle 4 =\)? Wait, maybe the sum of \(\angle 3\), \(\angle 4\), and the \(80^\circ\) angle is \(180^\circ\)? Wait, \(\angle 3 = 55^\circ\), \(80^\circ\), so \(55 + 80 + \angle 4 = 180\)? Then \(\angle 4 = 180 - 55 - 80 = 45^\circ\)? But \(m\angle 2 = 45^\circ\), which is vertical to \(\angle 4\)? Wait, yes! \(\angle 2\) and \(\angle 4\) are vertical angles, so they should be equal. Since \(m\angle 2 = 45^\circ\), then \(m\angle 4 = 45^\circ\). Alternatively, using the straight line: the sum of \(80^\circ\), \(\angle 4\), and \(\angle 5\) (which is equal to \(\angle 7 = 55^\circ\) because of parallel lines) should be \(180^\circ\). Wait, \(80 + 45 + 55 = 180\), yes! So \(80 + \angle 4 + 55 = 180\), so \(\angle 4 = 180 - 80 - 55 = 45^\circ\).

Step1: Recall vertical angles or linear pair

\(\angle 2\) and \(\angle 4\) are vertical angles, so they are equal. Given \(m\angle 2 = 45^\circ\), so \(m\angle 4 = 45^\circ\). Alternatively, using linear pair: \(80^\circ + \angle 4 + 55^\circ = 180^\circ\) (since \(55^\circ\) is from \(180 - 125\) and corresponding angle). Solving: \(\angle 4 = 180 - 80 - 55 = 45^\circ\).

Answer:

\(45\)