Question

1. use the diagram below to decide if the following statements are true or false. choose the justification from the list to justify your response.

a. linear pairs are supplementary
b. vertical angles are congruent
c. corresponding angles are congruent when they are formed by parallel lines
d. alternate interior angles are congruent when they are formed by parallel lines
e. same side interior angles are supplementary when they are formed by parallel lines
f. alternate exterior angles are congruent when they are formed by parallel lines
g. the statement is false

statement	justification

|∠4≅∠6|
|∠1≅∠8|
|∠1 and ∠5 are supplementary|
|∠2≅∠3|
|∠7 and ∠8 are supplementary|
|∠8≅∠4|

Explanation:

Step1: Recall angle - pair properties

Linear pairs of angles add up to 180 degrees, so they are supplementary. Vertical angles are always congruent. When two parallel lines are cut by a transversal, corresponding angles are congruent, alternate - interior angles are congruent, same - side interior angles are supplementary, and alternate - exterior angles are congruent.

Step2: Analyze each statement in the table

• $\angle4$ and $\angle6$ are alternate - interior angles. Since the lines are parallel, $\angle4\cong\angle6$. Justification: Alternate interior angles are congruent when lines are parallel.
• $\angle1$ and $\angle8$ are alternate - exterior angles. Since the lines are parallel, $\angle1\cong\angle8$. Justification: Alternate exterior angles are congruent when lines are parallel.
• $\angle1$ and $\angle5$ are corresponding angles. They are not supplementary. Justification: Corresponding angles are congruent when lines are parallel, not supplementary.
• $\angle2$ and $\angle3$ are vertical angles. So, $\angle2\cong\angle3$. Justification: Vertical angles are congruent.
• $\angle7$ and $\angle8$ are a linear pair. So, $\angle7$ and $\angle8$ are supplementary. Justification: Linear pairs are supplementary.
• $\angle8$ and $\angle4$ are not congruent. Justification: They are neither vertical, corresponding, alternate - interior, nor alternate - exterior angles.

Answer:

a) True. Justification: Alternate interior angles are congruent when lines are parallel.
b) True. Justification: Alternate exterior angles are congruent when lines are parallel.
c) False. Justification: Corresponding angles are congruent when lines are parallel, not supplementary.
d) True. Justification: Vertical angles are congruent.
e) True. Justification: Linear pairs are supplementary.
f) False. Justification: They are neither vertical, corresponding, alternate - interior, nor alternate - exterior angles.