Question

do you know how? use the diagram for exercises 5 - 8. classify each pair of angles. compare angle measures, and give the postulate or theorem that justifies it. 5. ∠2 and ∠6 6. ∠3 and ∠5 if m∠1 = 71, find the measure of each angle. 7. ∠5 8. ∠7 9. elm st. and spruce st. are parallel. what is m∠1?

Explanation:

Step1: Identify angle - pair relationships

For $\angle2$ and $\angle6$, they are corresponding angles. Corresponding angles are congruent when two parallel lines are cut by a transversal. The postulate is the Corresponding Angles Postulate.
For $\angle3$ and $\angle5$, they are alternate - interior angles. Alternate - interior angles are congruent when two parallel lines are cut by a transversal. The theorem is the Alternate - Interior Angles Theorem.

Step2: Find angle measures when $m\angle1 = 71^{\circ}$

$\angle1$ and $\angle5$ are corresponding angles. Since corresponding angles are congruent when two parallel lines are cut by a transversal, if $m\angle1 = 71^{\circ}$, then $m\angle5=71^{\circ}$.
$\angle1$ and $\angle7$ are vertical - angles with $\angle1$ and $\angle3$, and $\angle3$ and $\angle7$ are corresponding angles. Also, vertical angles are congruent. So $m\angle7 = 71^{\circ}$.

Step3: Solve for $m\angle1$ in the street - intersection problem

Elm St. and Spruce St. are parallel. The angle of $112^{\circ}$ and $\angle1$ are same - side interior angles. Same - side interior angles are supplementary when two parallel lines are cut by a transversal. Let $m\angle1=x$. Then $x + 112^{\circ}=180^{\circ}$, so $x=180^{\circ}-112^{\circ}=68^{\circ}$.

Answer:

1. Corresponding angles; congruent; Corresponding Angles Postulate
2. Alternate - interior angles; congruent; Alternate - Interior Angles Theorem
3. $m\angle5 = 71^{\circ}$
4. $m\angle7 = 71^{\circ}$
5. $m\angle1 = 68^{\circ}$