Question

directions: complete each proof.

1. given: (mangle4 + mangle7 = 180^{circ}) prove: (cparallel d)

statements reasons
(mangle4 + mangle7 = 180) given

1. given: (mangle3 = mangle8) prove: (mangle3 + mangle6 = 180^{circ})

statements reasons
(mangle3 + mangle6 = 180) def of supplementary

1. given: (pparallel q;angle1congangle5) prove: (angle2congangle5)

statements reasons

Explanation:

1.

Step1: Identify angle - pair relationship

$\angle4$ and $\angle7$ are same - side interior angles.

Step2: Apply parallel - line theorem

If same - side interior angles are supplementary, then the lines are parallel. Since $m\angle4 + m\angle7=180^{\circ}$, by the Converse of the Same - Side Interior Angles Postulate, $c\parallel d$.

Step1: Observe angle relationships

$\angle8$ and $\angle6$ are supplementary (linear pair), so $m\angle8 + m\angle6=180^{\circ}$.

Step2: Substitute using the given

Given $m\angle3 = m\angle8$. Substituting $m\angle3$ for $m\angle8$ in the equation $m\angle8 + m\angle6 = 180^{\circ}$, we get $m\angle3 + m\angle6=180^{\circ}$.

Step1: Use the property of parallel lines

Since $p\parallel q$, $\angle1\cong\angle3$ (corresponding angles postulate).

Step2: Use the given congruence

Given $\angle1\cong\angle5$.

Step3: Apply the transitive property

By the Transitive Property of Congruence, if $\angle1\cong\angle3$ and $\angle1\cong\angle5$, then $\angle3\cong\angle5$.

Step4: Use vertical - angle congruence

$\angle2\cong\angle3$ (vertical angles are congruent).

Step5: Apply the transitive property again

By the Transitive Property of Congruence, since $\angle2\cong\angle3$ and $\angle3\cong\angle5$, then $\angle2\cong\angle5$.

Answer:

Statements	Reasons
$c\parallel d$	Converse of the Same - Side Interior Angles Postulate

2.