Question

1. a. find m∠ceb. b. find m∠deb. c. solve for x. 6. use the diagrams 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

When two parallel lines are cut by a transversal, we have the following relationships:

• Corresponding angles are congruent.
• Alternate interior angles are congruent.
• Alternate exterior angles are congruent.
• Same - side interior angles are supplementary.
• Vertical angles are congruent.
• Linear pairs are supplementary.

Step2: Analyze each statement

a) $\angle4\cong\angle6$

$\angle4$ and $\angle6$ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are congruent. Justification: Alternate interior angles are congruent when they are formed by parallel lines.

b) $\angle1\cong\angle8$

$\angle1$ and $\angle8$ are alternate exterior angles. When two parallel lines are cut by a transversal, alternate exterior angles are congruent. Justification: Alternate exterior angles are congruent when they are formed by parallel lines.

c) $\angle1$ and $\angle5$ are supplementary

$\angle1$ and $\angle5$ are same - side interior angles. When two parallel lines are cut by a transversal, same - side interior angles are supplementary. Justification: Same side interior angles are supplementary when they are formed by parallel lines.

d) $\angle2\cong\angle3$

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

e) $\angle7$ and $\angle8$ are supplementary

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

f) $\angle8\cong\angle4$

$\angle8$ and $\angle4$ are corresponding angles. When two parallel lines are cut by a transversal, corresponding angles are congruent. Justification: Corresponding angles are congruent when they are formed by parallel lines.

Answer:

a) Justification: Alternate interior angles are congruent when they are formed by parallel lines.
b) Justification: Alternate exterior angles are congruent when they are formed by parallel lines.
c) Justification: Same side interior angles are supplementary when they are formed by parallel lines.
d) Justification: Vertical angles are congruent.
e) Justification: Linear pairs are supplementary.
f) Justification: Corresponding angles are congruent when they are formed by parallel lines.