Question

1. given the following information, determine which lines, if any, are parallel. state the converse that justifies your answer.

given	parallel lines	converse
b. m∠14 + m∠18 = 180°
c. ∠4≅∠20
d. ∠3≅∠16
e. ∠10≅∠12
f. m∠7 + m∠19 = 180°
g. ∠6≅∠17
h. ∠9≅∠24
i. ∠2≅∠21
j. m∠3 + m∠7 = 180°
k. ∠6≅∠11
l. ∠1≅∠3
m. ∠12≅∠15
n. m∠13 + m∠16 = 180°
o. ∠15≅∠18

Explanation:

Step1: Recall parallel - line converse theorems

If alternate interior angles are congruent, then the lines are parallel. If same - side interior angles are supplementary, then the lines are parallel. If corresponding angles are congruent, then the lines are parallel.

Step2: Analyze each given condition

a. $\angle10\cong\angle15$

$\angle10$ and $\angle15$ are alternate interior angles. The parallel lines are $l$ and $m$. The converse is: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

b. $m\angle14 + m\angle18=180^{\circ}$

$\angle14$ and $\angle18$ are same - side interior angles. The parallel lines are $k$ and $m$. The converse is: If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

c. $\angle4\cong\angle20$

$\angle4$ and $\angle20$ are corresponding angles. The parallel lines are $k$ and $l$. The converse is: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

d. $\angle3\cong\angle16$

$\angle3$ and $\angle16$ are alternate interior angles. The parallel lines are $k$ and $m$. The converse is: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

e. $\angle10\cong\angle12$

$\angle10$ and $\angle12$ are vertical angles, and this does not imply parallel lines. So, no parallel lines.

f. $m\angle7 + m\angle19 = 180^{\circ}$

$\angle7$ and $\angle19$ are same - side interior angles. The parallel lines are $k$ and $l$. The converse is: If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

g. $\angle6\cong\angle17$

$\angle6$ and $\angle17$ are corresponding angles. The parallel lines are $l$ and $m$. The converse is: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

h. $\angle9\cong\angle24$

$\angle9$ and $\angle24$ are corresponding angles. The parallel lines are $k$ and $m$. The converse is: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

i. $\angle2\cong\angle21$

$\angle2$ and $\angle21$ are corresponding angles. The parallel lines are $k$ and $l$. The converse is: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

j. $m\angle3 + m\angle7=180^{\circ}$

$\angle3$ and $\angle7$ are same - side interior angles. The parallel lines are $k$ and $l$. The converse is: If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

k. $\angle6\cong\angle11$

$\angle6$ and $\angle11$ are alternate interior angles. The parallel lines are $k$ and $l$. The converse is: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

l. $\angle1\cong\angle3$

$\angle1$ and $\angle3$ are vertical angles, and this does not imply parallel lines. So, no parallel lines.

m. $\angle12\cong\angle15$

$\angle12$ and $\angle15$ are corresponding angles. The parallel lines are $k$ and $m$. The converse is: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

n. $m\angle13 + m\angle16=180^{\circ}$

$\angle13$ and $\angle16$ are same - side interior angles. The parallel lines are $l$ and $m$. The converse is: If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.

o. $\angle15\cong\angle18$

$\angle15$ and $\angle18$ are vertical angles, and this does not imply parallel lines. So, no parallel lines.

Answer:

Given	Parallel Lines	Converse
b. $m\angle14 + m\angle18 = 180^{\circ}$	$k$ and $m$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.
c. $\angle4\cong\angle20$	$k$ and $l$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
d. $\angle3\cong\angle16$	$k$ and $m$	If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
e. $\angle10\cong\angle12$	None	-
f. $m\angle7 + m\angle19 = 180^{\circ}$	$k$ and $l$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.
g. $\angle6\cong\angle17$	$l$ and $m$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
h. $\angle9\cong\angle24$	$k$ and $m$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
i. $\angle2\cong\angle21$	$k$ and $l$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
j. $m\angle3 + m\angle7 = 180^{\circ}$	$k$ and $l$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.
k. $\angle6\cong\angle11$	$k$ and $l$	If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
l. $\angle1\cong\angle3$	None	-
m. $\angle12\cong\angle15$	$k$ and $m$	If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
n. $m\angle13 + m\angle16 = 180^{\circ}$	$l$ and $m$	If two lines are cut by a transversal and same - side interior angles are supplementary, then the lines are parallel.
o. $\angle15\cong\angle18$	None	-