Question

1. given: ∠2 and ∠3 are supplementary prove: r || m statements reasons a. ∠2 and ∠3 are supplementary a. given b. m∠2 + m∠3 = 180° b. c. m∠2 = m∠1 c. d. m∠1 + m∠3 = 180° d. e. e. 4) given: c || d, ∠2 ≅ ∠3 prove: a || b statements reasons a. c || d a. given b. ∠1 ≅ ∠3 b. c. ∠2 ≅ ∠3 c. d. ∠1 ≅ ∠2 d. e. a || b e.

Explanation:

Step1: Definition of supplementary angles

If two angles are supplementary, the sum of their measures is 180°. So, if ∠2 and ∠3 are supplementary, then $m\angle2 + m\angle3=180^{\circ}$.

Step2: Vertical - angle theorem

∠1 and ∠2 are vertical angles. Vertical angles are congruent, so $m\angle2 = m\angle1$.

Step3: Substitution property

Since $m\angle2 + m\angle3 = 180^{\circ}$ and $m\angle2 = m\angle1$, we can substitute $m\angle1$ for $m\angle2$ to get $m\angle1 + m\angle3=180^{\circ}$.

Step4: Converse of same - side interior angles postulate

If the sum of the measures of two same - side interior angles is 180°, then the two lines are parallel. Since $m\angle1 + m\angle3 = 180^{\circ}$, then $r\parallel m$.

For the second proof:

Step1: Given

$c\parallel d$ is given.

Step2: Corresponding angles postulate

When two parallel lines ($c$ and $d$) are cut by a transversal, corresponding angles are congruent. So, $\angle1\cong\angle3$.

Step3: Given

$\angle2\cong\angle3$ is given.

Step4: Transitive property of congruence

If $\angle1\cong\angle3$ and $\angle2\cong\angle3$, then $\angle1\cong\angle2$ by the transitive property of congruence.

Step5: Converse of corresponding angles postulate

If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Since $\angle1\cong\angle2$, then $a\parallel b$.

Answer:

For the first proof:

Statements	Reasons
b. $m\angle2 + m\angle3 = 180^{\circ}$	b. Definition of supplementary angles
c. $m\angle2 = m\angle1$	c. Vertical - angle theorem
d. $m\angle1 + m\angle3 = 180^{\circ}$	d. Substitution property
e. $r\parallel m$	e. Converse of same - side interior angles postulate

For the second proof:

Statements	Reasons
b. $\angle1\cong\angle3$	b. Corresponding angles postulate
c. $\angle2\cong\angle3$	c. Given
d. $\angle1\cong\angle2$	d. Transitive property of congruence
e. $a\parallel b$	e. Converse of corresponding angles postulate