Question

1. generalize in the diagram shown, if x + y = 180, label the remaining angles as x° or y°.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle vertical to the given $x^{\circ}$ angle is also $x^{\circ}$.

Step2: Use linear - pair property

Since $x + y=180$, the angle adjacent to the given $x^{\circ}$ angle (forming a linear - pair) is $y^{\circ}$.

Step3: Use corresponding - angle and alternate - angle properties

Corresponding angles and alternate interior/exterior angles formed by parallel lines (assuming the horizontal lines are parallel) are equal. So, the angles in the upper - part of the intersection that are corresponding or alternate to the already labeled angles will have the same measures. The angle corresponding to the given $x^{\circ}$ angle is $x^{\circ}$, and the angle corresponding to the $y^{\circ}$ angle (adjacent to the given $x^{\circ}$ angle) is $y^{\circ}$.

Answer:

The angle vertical to the given $x^{\circ}$ is $x^{\circ}$. The angle adjacent to the given $x^{\circ}$ (forming a linear - pair) is $y^{\circ}$. The corresponding and alternate angles to the $x^{\circ}$ angle are $x^{\circ}$, and the corresponding and alternate angles to the $y^{\circ}$ angle are $y^{\circ}$.