Question

given line x is parallel to line y and m∠2 = 23°. what is m∠7? question 8 in the figure m∠2=(20x + 52)° and m∠7=(40x - 48)°. for what value of x is m∥n?

Explanation:

Step1: Use vertical - angle property

Since vertical angles are equal, $\angle2$ and $\angle4$ are vertical angles, so $m\angle4 = m\angle2=23^{\circ}$.

Step2: Use corresponding - angle property for parallel lines

Because line $x$ is parallel to line $y$, $\angle4$ and $\angle7$ are alternate - exterior angles. For parallel lines, alternate - exterior angles are equal. So $m\angle7 = 23^{\circ}$.

For the second part:

Step1: Use the property of alternate - exterior angles for parallel lines

If $m\parallel n$, then $\angle2$ and $\angle7$ (alternate - exterior angles) are equal. Set up the equation $20x + 52=40x-48$.

Step2: Solve the equation for $x$

Subtract $20x$ from both sides: $52 = 40x-20x - 48$, which simplifies to $52=20x - 48$.
Add 48 to both sides: $52 + 48=20x$, so $100 = 20x$.
Divide both sides by 20: $x=\frac{100}{20}=5$.

Answer:

$m\angle7 = 23^{\circ}$
$x = 5$