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 = 46° m∠3 = \boxed{ }°

Explanation:

Step1: Identify angle relationships

Angles on a straight line sum to \(180^\circ\). Also, vertical angles are equal. We know \(m\angle1 = 80^\circ\) (vertical to the \(80^\circ\) angle) and \(m\angle2\) can be found, but for \(\angle3\), we use the fact that \(\angle1\), \(\angle2\), and \(\angle3\) form a straight line? Wait, no, actually, looking at the diagram, \(\angle3\), the \(80^\circ\) angle, and \(\angle1\) – wait, no, let's re-express. Wait, the \(80^\circ\) angle, \(\angle4\), and the triangle? Wait, no, lines \(m\) and \(n\) are parallel. Wait, maybe \(\angle3\) is equal to \(\angle9\) or something, but actually, let's check the straight line. Wait, the angle adjacent to \(125^\circ\) is \(180 - 125 = 55^\circ\)? No, wait, the \(125^\circ\) angle and \(\angle7\) are supplementary, so \(m\angle7 = 180 - 125 = 55^\circ\). Then, since \(m\) and \(n\) are parallel, the triangle's angle: wait, maybe \(\angle4\) is equal to \(\angle7\) (alternate interior angles)? So \(m\angle4 = 55^\circ\)? Wait, no, the \(80^\circ\) angle, \(\angle4\), and the angle in the triangle: wait, maybe I made a mistake. Wait, the problem gives \(m\angle1 = 80^\circ\) (wait, no, the user wrote \(m\angle1 = 80^\circ\)? Wait, the diagram has an \(80^\circ\) angle. Wait, maybe \(\angle3\), \(\angle1\), and \(\angle2\) – no, let's think again. Wait, the sum of angles around a point is \(360^\circ\), but on a straight line, it's \(180^\circ\). Wait, \(\angle1\) is \(80^\circ\) (vertical angle to the given \(80^\circ\)), \(\angle2\) – wait, maybe \(\angle3\) is equal to \(\angle4\)? No, wait, let's use the fact that \(\angle3\), the \(80^\circ\) angle, and \(\angle1\) – no, maybe \(\angle3\) is equal to \(m\angle2\)? Wait, no, the user has \(m\angle2 = 45^\circ\)? Wait, no, maybe I misread. Wait, the problem is to find \(m\angle3\). Let's see: angles on a straight line: \(\angle3 + 80^\circ + \angle1 = 180^\circ\)? Wait, \(\angle1\) is \(80^\circ\) (vertical angle), so \(\angle3 + 80^\circ + 80^\circ = 180^\circ\)? No, that can't be. Wait, maybe \(\angle3\) is equal to \(\angle7\) or something. Wait, the \(125^\circ\) angle and \(\angle7\) are supplementary, so \(m\angle7 = 55^\circ\). Then, since \(m\) and \(n\) are parallel, the alternate interior angle to \(\angle7\) is \(\angle4\), so \(m\angle4 = 55^\circ\). Then, the \(80^\circ\) angle, \(\angle4\), and \(\angle3\) – wait, no, \(\angle3\), \(\angle4\), and the \(80^\circ\) angle: wait, maybe \(\angle3\) is equal to \(180^\circ - 80^\circ - \angle4\)? Wait, \(\angle4 = 55^\circ\), so \(180 - 80 - 55 = 45^\circ\)? Wait, the user has \(m\angle2 = 45^\circ\), maybe \(\angle3\) is equal to \(\angle2\)? Wait, vertical angles? No, \(\angle3\) and \(\angle2\) – maybe \(\angle3 = 45^\circ\)? Wait, no, let's do it properly.

Wait, the key is: lines \(m\) and \(n\) are parallel. The angle adjacent to \(125^\circ\) is \(180 - 125 = 55^\circ\) (linear pair), so \(m\angle7 = 55^\circ\). Then, \(\angle7\) and \(\angle4\) are alternate interior angles (since \(m \parallel n\) and the transversal), so \(m\angle4 = 55^\circ\). Now, the \(80^\circ\) angle, \(\angle4\), and \(\angle3\) – wait, the angle at the intersection: \(\angle3 + 80^\circ + \angle4 = 180^\circ\) (since they form a straight line). So \(m\angle3 = 180 - 80 - 55 = 45^\circ\). Wait, but the user has \(m\angle2 = 45^\circ\), maybe \(\angle3\) is equal to \(\angle2\) (vertical angles or something). So \(m\angle3 = 45^\circ\).

Step1: Find supplementary angle to \(125^\circ\)

The angle adjacent to \(125^\circ\) (linear pair) is \(180^\circ - 125^\circ = 55^\circ\), so \(m\angle7 = 55^\circ\).

Step2: Alternate interior angles ( \(m \parallel n\) )

\(\angle7\) and \(\angle4\) are alternate interior angles, so \(m\angle4 = 55^\circ\).

Step3: Straight line angle sum

Angles \(80^\circ\), \(\angle4\), and \(\angle3\) are on a straight line, so \(m\angle3 + 80^\circ + 55^\circ = 180^\circ\).
Solving for \(m\angle3\): \(m\angle3 = 180 - 80 - 55 = 45^\circ\).

Answer:

\(45\)