Question

1. use the diagram below to decide if the following statements are true or false. choose one 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
a) ∠4≅∠6
b) ∠1≅∠8
c) ∠1 and ∠5 are supplementary
d) ∠2≅∠3
e) ∠7 and ∠8 are supplementary
f) ∠8≅∠4

Explanation:

Step1: Recall angle - pair relationships

Linear pairs add up to 180° (supplementary), vertical angles are congruent, corresponding, alternate - interior, and alternate - exterior angles are congruent when lines are parallel, and same - side interior angles are supplementary when lines are parallel.

Step2: Analyze each statement about the given angles

a)

$\angle4$ and $\angle6$ are alternate - interior angles. Since the lines are parallel, $\angle4\cong\angle6$. Justification: Alternate interior angles are congruent when they are formed by parallel lines.

b)

$\angle1$ and $\angle8$ are alternate - exterior angles. Since the lines are parallel, $\angle1\cong\angle8$. Justification: Alternate exterior angles are congruent when they are formed by parallel lines.

c)

$\angle1$ and $\angle5$ are corresponding angles. Corresponding angles are congruent when lines are parallel, not supplementary. Justification: The statement is false.

d)

$\angle2$ and $\angle3$ are vertical angles. Vertical angles are congruent. Justification: Vertical angles are congruent.

e)

$\angle7$ and $\angle8$ form a linear pair. Linear pairs are supplementary. Justification: Linear pairs are supplementary.

f)

$\angle8$ and $\angle4$ are not related by any of the special angle - pair relationships for parallel lines. Justification: The statement is false.

Answer:

Statement	Justification
b) $\angle1\cong\angle8$	Alternate exterior angles are congruent when they are formed by parallel lines
c) $\angle1$ and $\angle5$ are supplementary	The statement is false
d) $\angle2\cong\angle3$	Vertical angles are congruent
e) $\angle7$ and $\angle8$ are supplementary	Linear pairs are supplementary
f) $\angle8\cong\angle4$	The statement is false