Question

1. given a. ∠2≅∠4 b. ∠5≅∠10 c. m∠6 + m∠10 = 180° d. ∠1≅∠14 e. m∠14 + m∠15 = 180° f. ∠11≅∠16 g. ∠4≅∠15 h. ∠10≅∠12 i. m∠9 + m∠13 = 180° j. ∠2≅∠7 k. ∠6≅∠11

parallel lines
converse
c || d (corresponding ∠s)
a || b (consecutive inter ∠s)

Explanation:

Step1: Recall parallel - line postulates and theorems

If corresponding angles are congruent, alternate - interior angles are congruent, alternate - exterior angles are congruent, or consecutive interior angles are supplementary, then the lines are parallel.

Step2: Analyze each part

a. $\angle2\cong\angle4$

These are vertical angles, and vertical angles are always congruent. This does not prove any lines are parallel.

b. $\angle5\cong\angle10$

$\angle5$ and $\angle10$ are corresponding angles. By the converse of the corresponding - angles postulate, if corresponding angles are congruent, then the lines are parallel. So, $c\parallel d$.

c. $m\angle6 + m\angle10=180^{\circ}$

$\angle6$ and $\angle10$ are consecutive interior angles. By the converse of the consecutive - interior angles theorem, if consecutive interior angles are supplementary, then the lines are parallel. So, $a\parallel b$.

d. $\angle1\cong\angle14$

These angles have no special relationship (not corresponding, alternate - interior, alternate - exterior, or consecutive interior) that would prove lines are parallel.

e. $m\angle14 + m\angle15 = 180^{\circ}$

$\angle14$ and $\angle15$ are a linear pair (adjacent and supplementary), not a special angle - pair for parallel lines. This does not prove any lines are parallel.

f. $\angle11\cong\angle16$

$\angle11$ and $\angle16$ are corresponding angles. By the converse of the corresponding - angles postulate, $c\parallel d$.

g. $\angle4\cong\angle15$

These angles have no special relationship (not corresponding, alternate - interior, alternate - exterior, or consecutive interior) that would prove lines are parallel.

h. $\angle10\cong\angle12$

These are vertical angles, and vertical angles are always congruent. This does not prove any lines are parallel.

i. $m\angle9 + m\angle13=180^{\circ}$

$\angle9$ and $\angle13$ are consecutive interior angles. By the converse of the consecutive - interior angles theorem, $c\parallel d$.

j. $\angle2\cong\angle7$

These are alternate - exterior angles. By the converse of the alternate - exterior angles theorem, $a\parallel b$.

k. $\angle6\cong\angle11$

These are alternate - interior angles. By the converse of the alternate - interior angles theorem, $a\parallel b$.

Answer:

a. No parallel lines
b. $c\parallel d$, Converse of corresponding - angles postulate
c. $a\parallel b$, Converse of consecutive - interior angles theorem
d. No parallel lines
e. No parallel lines
f. $c\parallel d$, Converse of corresponding - angles postulate
g. No parallel lines
h. No parallel lines
i. $c\parallel d$, Converse of consecutive - interior angles theorem
j. $a\parallel b$, Converse of alternate - exterior angles theorem
k. $a\parallel b$, Converse of alternate - interior angles theorem