Question

1. given the diagram below, find x and z.

a. solve for z.
b. find m∠ceb.
c. find m∠deb.
d. solve for x.

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

Explanation:

Step1: Solve for z using vertical - angle property

Vertical angles are equal. So, \(5z - 16=3z + 32\).
Subtract \(3z\) from both sides: \(5z-3z - 16=3z-3z + 32\), which simplifies to \(2z-16 = 32\).
Add 16 to both sides: \(2z-16 + 16=32 + 16\), getting \(2z=48\).
Divide both sides by 2: \(z=\frac{48}{2}=24\).

Step2: Solve for x using linear - pair property

The angles \((9x + 6)\) and \((3z + 32)\) form a linear - pair, so \((9x + 6)+(3z + 32)=180\).
Substitute \(z = 24\) into the equation: \((9x + 6)+(3\times24+32)=180\).
First, calculate \(3\times24+32=72 + 32=104\).
The equation becomes \(9x+6 + 104=180\), or \(9x+110 = 180\).
Subtract 110 from both sides: \(9x+110 - 110=180 - 110\), getting \(9x=70\).
Divide both sides by 9: \(x=\frac{70}{9}\approx7.78\).

Step3: Find \(m\angle CEB\)

\(m\angle CEB=3z + 32\). Substitute \(z = 24\), then \(m\angle CEB=3\times24+32=72 + 32 = 104^{\circ}\).

Step4: Find \(m\angle DEB\)

Since \(\angle CEB\) and \(\angle DEB\) form a linear - pair, \(m\angle DEB=180 - m\angle CEB\).
\(m\angle DEB=180 - 104=76^{\circ}\).

Step5: Answer question 6

a. True. By definition, linear pairs of angles are supplementary (their sum is \(180^{\circ}\)).
b. True. Vertical angles are always congruent.
c. True. When two parallel lines are cut by a transversal, corresponding angles are congruent.
d. True. When two parallel lines are cut by a transversal, alternate interior angles are congruent.
e. True. When two parallel lines are cut by a transversal, same - side interior angles are supplementary.
f. True. When two parallel lines are cut by a transversal, alternate exterior angles are congruent.
g. There is no statement given for 'g' in the problem description, so we assume it's an error in the problem - set up.

Answer:

a. \(z = 24\), \(x=\frac{70}{9}\approx7.78\)
b. \(m\angle CEB = 104^{\circ}\)
c. \(m\angle DEB = 76^{\circ}\)
d. \(x=\frac{70}{9}\approx7.78\)
6.
a. True, linear - pair definition.
b. True, vertical - angle property.
c. True, parallel - line property.
d. True, parallel - line property.
e. True, parallel - line property.
f. True, parallel - line property.
g. No valid statement provided.