Question

1. determine what lines can be parallel based on the given information. state the theorem that justifies.

a. ∠2≅∠3
b. ∠1≅∠4
c. m∠5 + m∠8 = 180°
d. ∠6≅∠7

Explanation:

Step1: Analyze $\angle2\cong\angle3$

$\angle2$ and $\angle3$ are alternate - interior angles. By the Alternate - Interior Angles Theorem, if alternate - interior angles are congruent, then the lines are parallel. So, $p\parallel s$.

Step2: Analyze $\angle1\cong\angle4$

$\angle1$ and $\angle4$ are corresponding angles. By the Corresponding Angles Theorem, if corresponding angles are congruent, then the lines are parallel. So, $p\parallel s$.

Step3: Analyze $m\angle5 + m\angle8=180^{\circ}$

$\angle5$ and $\angle8$ are same - side interior angles. By the Same - Side Interior Angles Postulate, if same - side interior angles are supplementary, then the lines are parallel. So, $p\parallel s$.

Step4: Analyze $\angle6\cong\angle7$

$\angle6$ and $\angle7$ are alternate - exterior angles. By the Alternate - Exterior Angles Theorem, if alternate - exterior angles are congruent, then the lines are parallel. So, $p\parallel s$.

Answer:

a. Lines $p$ and $s$ are parallel by the Alternate - Interior Angles Theorem.
b. Lines $p$ and $s$ are parallel by the Corresponding Angles Theorem.
c. Lines $p$ and $s$ are parallel by the Same - Side Interior Angles Postulate.
d. Lines $p$ and $s$ are parallel by the Alternate - Exterior Angles Theorem.