Question

a corner of a rectangle is cut, creating a trapezoid. what is the value of x? 105° 115° 125° 135°

Explanation:

Step1: Recall rectangle angle property

A rectangle has all angles equal to \(90^\circ\). When a corner is cut, we form a triangle with a \(45^\circ\) angle and the original right angle.

Step2: Use linear pair or supplementary angles

The sum of angles on a straight line is \(180^\circ\). The angle adjacent to \(x\) and the \(45^\circ\) angle, along with the right angle, but more simply, the angle \(x\) and the angle formed by the cut (which is supplementary to the angle in the triangle) relate. Wait, actually, when we cut the rectangle's corner (a right angle, \(90^\circ\)) to form a triangle with a \(45^\circ\) angle, the angle adjacent to \(x\) inside the trapezoid: the triangle is isosceles? Wait, no. The original angle is \(90^\circ\), we cut it, so the two angles formed with the cut line: one is \(45^\circ\) (given), and the other angle in the triangle is \(180^\circ - 90^\circ - 45^\circ= 45^\circ\)? Wait, no. Wait, the rectangle's angle is \(90^\circ\), when we cut it, we have a straight line (the side of the trapezoid) and the cut line. So the angle \(x\) and the angle adjacent to it (from the cut) should add up to \(180^\circ\)? Wait, no. Wait, the correct approach: in a rectangle, each angle is \(90^\circ\). When we cut a corner, we create a triangle where one angle is \(90^\circ\) (the rectangle's angle), but wait, no—when we cut the corner, we remove a triangle, so the remaining angle at that vertex of the trapezoid: the sum of angles around a point? No, along the side. Wait, the two angles formed by the cut on the rectangle's side: one is \(45^\circ\) (given), and the other angle (let's call it \(y\)) and \(x\) are supplementary (since they are on a straight line). But also, in the rectangle, the angle was \(90^\circ\), so the triangle that was cut off has angles: the right angle was \(90^\circ\), so the two angles of the triangle are \(45^\circ\) and \(45^\circ\) (since \(180 - 90 - 45 = 45\)). Wait, no, the triangle is a right triangle? Wait, no, the rectangle's corner is a right angle, so when we cut it with a line, we form a triangle with one angle \(45^\circ\), so the other angle in the triangle is \(90^\circ - 45^\circ = 45^\circ\)? No, that's not right. Wait, the correct way: the angle \(x\) and the \(45^\circ\) angle are supplementary to the right angle? Wait, no. Let's think again. The original angle is \(90^\circ\). When we cut the corner, we have a straight line (the edge of the trapezoid) and the cut line. So the angle between the trapezoid's side and the cut line is \(45^\circ\), so the angle \(x\) and the angle adjacent to it (from the cut) should add up to \(180^\circ\). But the adjacent angle: since the rectangle's angle is \(90^\circ\), and we have a triangle with angle \(45^\circ\), the angle between the trapezoid's side and the cut line is \(180^\circ - 90^\circ - 45^\circ = 45^\circ\)? No, that's not. Wait, I'm overcomplicating. The correct formula: in a trapezoid, consecutive angles between the two parallel sides are supplementary? Wait, no, this is a trapezoid formed by cutting a rectangle, so it's a right trapezoid? Wait, the original rectangle has two right angles at the top, and when we cut the bottom right corner, we have a trapezoid with two right angles at the top, one angle \(x\) at the bottom left, and the angle at the bottom right: the cut creates a \(45^\circ\) angle. Wait, the sum of angles in a quadrilateral is \(360^\circ\). The trapezoid has two right angles (\(90^\circ\) each), one angle \(x\), and one angle which is \(180^\circ - 45^\circ = 135^\circ\)? Wait, no. Wait, let's list the angles: two right angles (\(90^\circ\)), one angle \(x\), and the fourth angle: when we cut the rectangle's corner (a \(90^\circ\) angle), we replace it with an angle formed by the cut. The cut line makes a \(45^\circ\) angle with the side, so the angle inside the trapezoid at that corner is \(180^\circ - 45^\circ = 135^\circ\)? Wait, no. Wait, the rectangle's side is straight, so the angle between the cut line and the side is \(45^\circ\), so the angle adjacent to \(x\) (on the same side) would be \(180^\circ - 45^\circ = 135^\circ\)? Wait, no, let's use the fact that in the rectangle, the angle is \(90^\circ\), so when we cut it, the two angles formed by the cut line with the rectangle's sides: one is \(45^\circ\) (given), and the other angle (the one inside the trapezoid) is \(180^\circ - 45^\circ = 135^\circ\)? Wait, no, that's not. Wait, the correct answer is that \(x = 180^\circ - 45^\circ = 135^\circ\)? Wait, no, wait. Wait, the rectangle's angle is \(90^\circ\), we cut it, so the triangle has angles: \(90^\circ\) (the rectangle's angle), \(45^\circ\) (given), so the third angle is \(45^\circ\). Then, the angle \(x\) and the \(45^\circ\) angle (the one in the triangle) are supplementary? Wait, no. Wait, the side of the trapezoid and the cut line form a linear pair with \(x\)? No, I think I made a mistake. Let's start over.

The rectangle has four right angles (\(90^\circ\)). When we cut a corner, we remove a triangle. The triangle has one angle equal to the angle of the cut with the side, which is \(45^\circ\), and the other angle in the triangle: since the rectangle's angle is \(90^\circ\), the triangle is a right triangle? Wait, no—when you cut the corner of a rectangle, you are cutting along a line, so the triangle removed has two angles: one is the angle between the cut line and the side ( \(45^\circ\) ), and the other is the angle between the cut line and the other side (let's call it \(y\) ). Since the original angle is \(90^\circ\), we have \(45^\circ + y + 90^\circ = 180^\circ\) (sum of angles in a triangle)? No, a triangle's angles sum to \(180^\circ\), so if one angle is \(90^\circ\) (the rectangle's angle), then \(45^\circ + y + 90^\circ = 180^\circ\) → \(y = 45^\circ\). So the triangle is isosceles with two \(45^\circ\) angles. Then, the angle \(x\) is adjacent to the \(45^\circ\) angle ( \(y = 45^\circ\) ) and forms a linear pair, so \(x + 45^\circ = 180^\circ\) → \(x = 135^\circ\).

Step3: Calculate x

Using the linear pair (supplementary angles) concept: \(x + 45^\circ = 180^\circ\) (since they are on a straight line). Solving for \(x\): \(x = 180^\circ - 45^\circ = 135^\circ\).

Answer:

\(135^\circ\) (corresponding to the option 135°)