Question

1. review: geometry use the relationships in the diagram at right to solve for x and y. justify your solutions.

Explanation:

Step1: Use the property of corresponding angles

Since the two lines are parallel, the angle corresponding to the $61^{\circ}$ angle and the angle formed by $x$ and $43^{\circ}$ are related. The angle corresponding to the $61^{\circ}$ angle and the angle adjacent to $x$ (let's call it $a$) are equal, so $a = 61^{\circ}$.

Step2: Find the value of $x$

We know that the sum of angles on a straight - line is $180^{\circ}$. So $x+43^{\circ}+a = 180^{\circ}$. Substituting $a = 61^{\circ}$ into the equation, we get $x+43^{\circ}+61^{\circ}=180^{\circ}$, then $x=180^{\circ}-(43^{\circ}+61^{\circ})=76^{\circ}$.

Step3: Use the property of alternate - interior angles to find $y$

The angle $y$ and the angle formed by $x$ and $43^{\circ}$ are alternate - interior angles. The angle formed by $x$ and $43^{\circ}$ is $43^{\circ}+x = 43^{\circ}+76^{\circ}=119^{\circ}$. Since alternate - interior angles are equal for parallel lines, $y = 119^{\circ}$.

Answer:

$x = 76^{\circ}$, $y = 119^{\circ}$