Question

1. given\tparallel lines\tconverse

a. ∠9≅∠22
b. m∠8 + m∠13 = 180°
c. ∠1≅∠17
d. ∠16≅∠20
e. ∠5≅∠24
f. m∠4 + m∠17 = 180°
g. ∠10≅∠13
h. ∠3≅∠22
i. ∠5≅∠15
j. m∠11 + m∠16 = 180°
proofs: complete the proof below by filling in the missing reasons.

1. given: ∠4 and ∠5 are supplementary

prove: j || k
stations\treasons

1. ∠4 and ∠5 are supplementary\t1.
2. m∠4 + m∠5 = 180°\t2.
3. j || k\t3.
4. given: ∠1 and ∠2 form a linear pair; ∠1 and ∠4 are supplementary

prove: a || b
stations\treasons

1. ∠1 and ∠2 form a linear pair\t1.
2. ∠1 and ∠2 are supplementary\t2.
3. ∠1 and ∠4 are supplementary\t3.
4. ∠2≅∠4\t4.
5. a || b\t5.

Explanation:

Problem 2:

Step1: Identify angle - pair relationships

Use knowledge of corresponding, alternate - interior, alternate - exterior, and same - side (interior/exterior) angle relationships.

Step2: Apply parallel - line converse theorems

For example, if alternate - exterior angles are congruent, the lines are parallel.

Problem 3:

Step1: Recall supplementary - angle definition

Since $\angle4$ and $\angle5$ are supplementary, their measures add up to $180^{\circ}$.

Step2: Apply parallel - line theorem

Same - side interior angles being supplementary implies parallel lines.

Problem 4:

Step1: Use linear - pair property

Linear pairs of angles are supplementary.

Step2: Use supplement congruence property

If two angles are supplementary to the same angle, they are congruent.

Step3: Apply parallel - line theorem

Congruent corresponding angles imply parallel lines.

Answer:

Problem 2:

Given	Parallel Lines	Converse
b. $m\angle8 + m\angle13=180^{\circ}$	$p\parallel q$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.
c. $\angle1\cong\angle17$	$p\parallel r$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
d. $\angle16\cong\angle20$	$q\parallel r$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
e. $\angle5\cong\angle24$	$p\parallel r$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
f. $m\angle4 + m\angle17=180^{\circ}$	$p\parallel r$	If two lines are cut by a transversal and same - side exterior angles are supplementary, then the lines are parallel.
g. $\angle10\cong\angle13$	$p\parallel q$	If two lines are cut by a transversal and alternate - interior angles are congruent, then the lines are parallel.
h. $\angle3\cong\angle22$	$p\parallel r$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
i. $\angle5\cong\angle15$	$q\parallel r$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
j. $m\angle11 + m\angle16=180^{\circ}$	$p\parallel q$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

Problem 3:

Statements	Reasons
2. $m\angle4 + m\angle5=180^{\circ}$	Definition of supplementary angles
3. $j\parallel k$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

Problem 4:

Statements	Reasons
2. $\angle1$ and $\angle2$ are supplementary	Linear pairs are supplementary
3. $\angle1$ and $\angle4$ are supplementary	Given
4. $\angle2\cong\angle4$	Supplements of the same angle are congruent
5. $a\parallel b$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel