Question

name 2 - 2 additional practice proving lines parallel use the figure for exercises 1 - 4. using the given information, which lines can you conclude are parallel? state the theorem or postulate that justifies each answer. 1. ∠1≅∠4 2. ∠2≅∠3 3. ∠6≅∠7 4. m∠5 + m∠8 = 180

Explanation:

Step1: Identify angle - pair relationships

For $\angle1\cong\angle4$, they are vertical angles. Vertical - angle congruence does not directly prove parallel lines.

Step2: Analyze $\angle2\cong\angle3$

$\angle2$ and $\angle3$ are alternate interior angles. By the Alternate Interior Angles Theorem, if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel. So, we can conclude that the lines are parallel.

Step3: Analyze $\angle6\cong\angle7$

$\angle6$ and $\angle7$ are alternate exterior angles. By the Alternate Exterior Angles Theorem, if two lines are cut by a transversal and the alternate exterior angles are congruent, then the two lines are parallel.

Step4: Analyze $m\angle5 + m\angle8=180$

$\angle5$ and $\angle8$ are same - side exterior angles. By the Same - Side Exterior Angles Theorem, if two lines are cut by a transversal and the same - side exterior angles are supplementary, then the two lines are parallel.

Answer:

1. No theorem for parallel lines (vertical angles).
2. Alternate Interior Angles Theorem.
3. Alternate Exterior Angles Theorem.
4. Same - Side Exterior Angles Theorem.