Question

1. a. find m∠ceb. c. find m∠deb. d. solve for x. 6. 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: Identify angle - relationships

For the first part with angles \((10x + 47)^{\circ}\), \((15x+37)^{\circ}\) and \((10x - 13)^{\circ}\), we know that the sum of angles around a point is \(360^{\circ}\). Also, linear - pair angles are supplementary (sum to \(180^{\circ}\)).

Step2: Set up an equation to solve for \(x\)

Since \((10x + 47)+(15x + 37)+(10x-13)=360\) (sum of angles around a point).
Combine like - terms: \((10x+15x + 10x)+(47 + 37-13)=360\).
\(35x+71 = 360\).
Subtract 71 from both sides: \(35x=360 - 71=289\).
\(x=\frac{289}{35}\approx8.26\).

For the second part (true - false statements about angle relationships):
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. So this statement is True. 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. So this statement is True. Justification: Alternate exterior angles are congruent when they are formed by parallel lines.
c. \(\angle1\) and \(\angle5\) are supplementary:
\(\angle1\) and \(\angle5\) are corresponding angles. Corresponding angles are congruent when two parallel lines are cut by a transversal, not supplementary. So this statement is False. Justification: The statement is false.
d. \(\angle2\cong\angle3\):
\(\angle2\) and \(\angle3\) are vertical angles. Vertical angles are congruent. So this statement is True. Justification: Vertical angles are congruent.
e. \(\angle7\) and \(\angle8\) are supplementary:
\(\angle7\) and \(\angle8\) form a linear pair. Linear pairs are supplementary. So this statement is True. 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. So this statement is True. Justification: Corresponding angles are congruent when they are formed by parallel lines.

Answer:

d. \(x=\frac{289}{35}\approx8.26\)
a. True, Alternate interior angles are congruent when they are formed by parallel lines
b. True, Alternate exterior angles are congruent when they are formed by parallel lines
c. False, The statement is false
d. True, Vertical angles are congruent
e. True, Linear pairs are supplementary
f. True, Corresponding angles are congruent when they are formed by parallel lines