Question

the figure below has two pairs of parallel lines. find the measures of \\(\angle x\\) and \\(\angle y\\).

\\(m\angle x = \\)

Explanation:

Identify parallel lines and transversals

The figure shows two pairs of parallel lines:

1. One pair of horizontal lines indicated by single arrows.
2. One pair of slanted lines indicated by double arrows.

Find the measure of angle x

Using the pair of parallel lines with double arrows:

• The transversal is the horizontal line with single arrows.
• The angle measuring \(138^\circ\) and the angle \(x\) are consecutive interior angles.
• Therefore, they are supplementary:

$$138^\circ + x = 180^\circ$$

$$x = 180^\circ - 138^\circ = 42^\circ$$

Find the measure of angle y

Consider the triangle formed on the right:

• The top angle of this triangle is given as \(56^\circ\).
• The bottom-left interior angle of this triangle is \(x\), which is \(42^\circ\).
• The bottom-right interior angle of this triangle is \(y\).
• The sum of angles in a triangle is \(180^\circ\):

$$x + y + 56^\circ = 180^\circ$$

$$42^\circ + y + 56^\circ = 180^\circ$$

$$y + 98^\circ = 180^\circ$$

$$y = 180^\circ - 98^\circ = 82^\circ$$

Answer:

The figure below has two pairs of parallel lines. Find the measures of \(\angle x\) and \(\angle y\).

• \(m\angle x =\) <blank>\(42^\circ\)</blank>
• \(m\angle y =\) <blank>\(82^\circ\)</blank>