Question

in the figure below, lines m and n are perpendicular.
what is the measure of angle x?

Explanation:

Step1: Recall perpendicular lines property

Perpendicular lines form \(90^\circ\) angles. Also, vertical angles are equal, and we can use angle sum for right angles.

Step2: Calculate \(x\)

Since lines \(m\) and \(n\) are perpendicular, the angle between them is \(90^\circ\). The angle adjacent to \(49^\circ\) and \(x\) (in the right angle) should satisfy \(49^\circ + x = 90^\circ\)? Wait, no, actually, looking at the vertical angles or the right angle. Wait, the angle between \(m\) (vertical line) and the slant line is \(49^\circ\), and \(x\) is equal to the angle that, with \(49^\circ\), makes \(90^\circ\)? Wait, no, actually, since \(m\perp n\), the angle between \(m\) (vertical) and \(n\) (horizontal) is \(90^\circ\). The slant line creates a \(49^\circ\) angle with \(m\), so the angle between the slant line and \(n\) is \(90 - 49 = 41^\circ\)? Wait, no, \(x\) is vertical angle to the angle that is \(90 - 49\)? Wait, no, let's see: the angle between \(m\) (upward vertical) and the slant line is \(49^\circ\), so the angle between the slant line and the downward vertical (opposite of \(m\))? Wait, no, \(x\) is at the intersection, with the horizontal and vertical lines. Wait, the right angle is between \(m\) (vertical) and \(n\) (horizontal). The slant line crosses through, making a \(49^\circ\) angle with \(m\) (upward). So the angle between the slant line and \(n\) (horizontal) is \(90 - 49 = 41^\circ\)? No, wait, \(x\) is equal to the angle that is complementary to \(49^\circ\) because of the right angle. Wait, actually, since \(m\) and \(n\) are perpendicular, the angle between them is \(90^\circ\). The two angles formed by the slant line with \(m\) and \(n\) should add up to \(90^\circ\). So if one is \(49^\circ\), the other is \(90 - 49 = 41^\circ\)? Wait, no, \(x\) is vertical angle to that \(41^\circ\)? Wait, no, vertical angles are equal. Wait, the angle opposite to \(x\) is equal to \(49^\circ\)? No, wait, let's draw mentally: \(m\) is vertical (up and down), \(n\) is horizontal (left and right). They are perpendicular, so angle between them is \(90^\circ\). A slant line goes through the intersection, making a \(49^\circ\) angle with the upward \(m\) (so between upward \(m\) and slant line is \(49^\circ\)). Then, the angle between the slant line and the downward \(m\) (opposite direction) would be... Wait, no, \(x\) is at the bottom left, between the left horizontal, downward vertical, and the slant line. Wait, the angle between the slant line and the upward \(m\) is \(49^\circ\), so the angle between the slant line and the downward \(m\) is \(180 - 49 = 131\)? No, that's not. Wait, no, the sum of angles around a point is \(360^\circ\), but we have right angles. Wait, \(m\perp n\), so four right angles (each \(90^\circ\)) at the intersection. The slant line divides one right angle into \(49^\circ\) and \(90 - 49 = 41^\circ\). Then, \(x\) is equal to \(41^\circ\)? Wait, no, \(x\) is vertical angle to the \(41^\circ\) angle? Wait, no, vertical angles are equal. Wait, the angle between the slant line and \(n\) (horizontal) is \(41^\circ\) (since \(90 - 49 = 41\)), and \(x\) is equal to that angle because they are vertical angles. Wait, yes! Because the angle between the slant line and the horizontal \(n\) (right side) is \(41^\circ\), so the angle on the left side ( \(x\)) is also \(41^\circ\)? Wait, no, \(x\) is at the bottom left, between the left horizontal, downward vertical, and the slant line. Wait, the upward \(m\) and slant line: \(49^\circ\), so downward \(m\) and slant line: \(180 - 49 = 131\)? No, that's not. Wait, I think I made a mistake. Let's use the fact that \(m\perp n\), so \(\angle\) between \(m\) and \(n\) is \(90^\circ\). The slant line creates two angles with \(m\) and \(n\): \(\angle1 = 49^\circ\) (between \(m\) and slant line), \(\angle2 = 90^\circ - 49^\circ = 41^\circ\) (between slant line and \(n\)). Then, \(x\) is equal to \(\angle2\) because they are vertical angles. Wait, yes! So \(x = 41^\circ\)? Wait, no, wait, \(x\) is at the bottom, between the left horizontal, downward vertical, and the slant line. The angle between the slant line and the downward vertical (opposite of \(m\)) would be \(49^\circ\) (vertical angle to the \(49^\circ\) at the top). Wait, no, vertical angles are equal. The angle at the top between \(m\) (up) and slant line is \(49^\circ\), so the angle at the bottom between \(m\) (down) and slant line is also \(49^\circ\)? No, that's not. Wait, vertical angles are opposite each other. So the angle at the top (between up \(m\) and slant line) is \(49^\circ\), the angle at the bottom (between down \(m\) and slant line) is also \(49^\circ\) (vertical angles). Then, the angle between down \(m\) and left \(n\) is \(90^\circ\) (since \(m\perp n\)). So \(x\) is the angle between left \(n\) and slant line, which is \(90^\circ - 49^\circ = 41^\circ\). Yes, that makes sense. So \(x = 90 - 49 = 41\).

Answer:

\(41\)