Question

if m∠1 = 71, find the measure of each angle.

1. ∠5
2. ∠7

if m∠3 = 3x + 12 and m∠5 = 2x + 3, find the measure of each angle.

1. ∠5 10. ∠7
2. elm st. and spruce st. are parallel. what is m∠1?

Explanation:

Step1: Assume parallel - line properties

If we assume the lines are parallel, then corresponding angles are equal and vertical angles are equal. If $\angle1$ and $\angle5$ are corresponding angles (assuming parallel lines), then $m\angle5=m\angle1$.
$m\angle5 = 71$

Step2: Find $m\angle7$

$\angle5$ and $\angle7$ are vertical angles. Vertical angles are equal. So $m\angle7=m\angle5$.
$m\angle7 = 71$

Step3: Solve for $x$ when $m\angle3 = 3x + 12$ and $m\angle5=2x + 3$

If $\angle3$ and $\angle5$ are same - side interior angles (assuming parallel lines), then $m\angle3+m\angle5 = 180$ (same - side interior angles are supplementary).
$3x + 12+2x + 3=180$
$5x+15 = 180$
$5x=180 - 15$
$5x=165$
$x = 33$

Step4: Find $m\angle5$

Substitute $x = 33$ into the expression for $m\angle5$.
$m\angle5=2x + 3=2\times33+3=66 + 3=69$

Step5: Find $m\angle7$

Since $\angle5$ and $\angle7$ are vertical angles, $m\angle7=m\angle5 = 69$

Step6: Find $m\angle1$ in the street - intersection problem

If Elm St. and Spruce St. are parallel, and the given $112^{\circ}$ angle and $\angle1$ are corresponding angles.
$m\angle1 = 112$

Answer:

1. $m\angle5 = 71$
2. $m\angle7 = 71$
3. $m\angle5 = 69$
4. $m\angle7 = 69$
5. $m\angle1 = 112$