Question

given the following information, determine which lines, if any, are parallel. state the converse that justifies your answer.
given parallel lines converse
a. ∠10≅∠15
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, the lines are parallel. If same - side interior angles are supplementary, the lines are parallel. If corresponding angles are congruent, the lines are parallel.

Step2: Analyze each given condition

a. $\angle10\cong\angle15$

$\angle10$ and $\angle15$ are alternate interior angles. So, $l\parallel k$ by the Converse of the Alternate Interior Angles Theorem.

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

$\angle14$ and $\angle18$ are same - side interior angles. So, $l\parallel m$ by the Converse of the Same - Side Interior Angles Theorem.

c. $\angle4\cong\angle20$

$\angle4$ and $\angle20$ are corresponding angles. So, $j\parallel n$ by the Converse of the Corresponding Angles Theorem.

d. $\angle3\cong\angle16$

$\angle3$ and $\angle16$ are corresponding angles. So, $j\parallel k$ by the Converse of the Corresponding Angles Theorem.

e. $\angle10\cong\angle12$

$\angle10$ and $\angle12$ are alternate interior angles. So, $k\parallel m$ by the Converse of the Alternate Interior Angles Theorem.

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

$\angle7$ and $\angle19$ are same - side interior angles. So, $j\parallel m$ by the Converse of the Same - Side Interior Angles Theorem.

g. $\angle6\cong\angle17$

$\angle6$ and $\angle17$ are corresponding angles. So, $l\parallel n$ by the Converse of the Corresponding Angles Theorem.

h. $\angle9\cong\angle24$

$\angle9$ and $\angle24$ are corresponding angles. So, $l\parallel n$ by the Converse of the Corresponding Angles Theorem.

i. $\angle2\cong\angle21$

$\angle2$ and $\angle21$ are corresponding angles. So, $l\parallel n$ by the Converse of the Corresponding Angles Theorem.

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

$\angle3$ and $\angle7$ are same - side interior angles. So, $j\parallel l$ by the Converse of the Same - Side Interior Angles Theorem.

k. $\angle6\cong\angle11$

$\angle6$ and $\angle11$ are corresponding angles. So, $l\parallel k$ by the Converse of the Corresponding Angles Theorem.

l. $\angle1\cong\angle3$

$\angle1$ and $\angle3$ are vertical angles, and this does not imply parallel lines.

m. $\angle12\cong\angle15$

$\angle12$ and $\angle15$ are corresponding angles. So, $k\parallel m$ by the Converse of the Corresponding Angles Theorem.

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

$\angle13$ and $\angle16$ are same - side interior angles. So, $l\parallel m$ by the Converse of the Same - Side Interior Angles Theorem.

o. $\angle15\cong\angle18$

$\angle15$ and $\angle18$ are vertical angles, and this does not imply parallel lines.

Answer:

a. $l\parallel k$, Converse of Alternate Interior Angles Theorem
b. $l\parallel m$, Converse of Same - Side Interior Angles Theorem
c. $j\parallel n$, Converse of Corresponding Angles Theorem
d. $j\parallel k$, Converse of Corresponding Angles Theorem
e. $k\parallel m$, Converse of Alternate Interior Angles Theorem
f. $j\parallel m$, Converse of Same - Side Interior Angles Theorem
g. $l\parallel n$, Converse of Corresponding Angles Theorem
h. $l\parallel n$, Converse of Corresponding Angles Theorem
i. $l\parallel n$, Converse of Corresponding Angles Theorem
j. $j\parallel l$, Converse of Same - Side Interior Angles Theorem
k. $l\parallel k$, Converse of Corresponding Angles Theorem
l. No parallel lines
m. $k\parallel m$, Converse of Corresponding Angles Theorem
n. $l\parallel m$, Converse of Same - Side Interior Angles Theorem
o. No parallel lines