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
© gina wilson (all things algebra) llc, 201

Question

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°

Accepted Answer

# Explanation:
## Step1: Analyze ∠10 ≅ ∠15 (Part a)
∠10 and ∠15 are alternate exterior angles. By the Converse of Alternate Exterior Angles Theorem, if alternate exterior angles are congruent, the lines cut by the transversal are parallel. So the parallel lines would be the ones cut by the transversal creating ∠10 and ∠15, and the converse is "If alternate exterior angles are congruent, then the lines are parallel".
## Step2: Analyze $ m\angle14 + m\angle18 = 180^\circ $ (Part b)
∠14 and ∠18 are same - side interior angles. By the Converse of Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠14 and ∠18, and the converse is "If same - side interior angles are supplementary, then the lines are parallel".
## Step3: Analyze ∠4 ≅ ∠20 (Part c)
∠4 and ∠20 are corresponding angles. By the Converse of Corresponding Angles Theorem, if corresponding angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal creating ∠4 and ∠20, and the converse is "If corresponding angles are congruent, then the lines are parallel".
## Step4: Analyze ∠3 ≅ ∠16 (Part d)
∠3 and ∠16 are alternate interior angles. By the Converse of Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠3 and ∠16, and the converse is "If alternate interior angles are congruent, then the lines are parallel".
## Step5: Analyze ∠10 ≅ ∠12 (Part e)
∠10 and ∠12 are vertical angles? No, wait, if we consider the lines, ∠10 and ∠12: if they are alternate interior or corresponding? Wait, maybe ∠10 and ∠12 are formed by a transversal and two lines. If ∠10 ≅ ∠12, by the Converse of Alternate Interior Angles (if they are alternate interior) or Corresponding (if corresponding). Wait, maybe they are alternate interior angles, so the parallel lines are the ones cut by the transversal, and the converse is "If alternate interior angles are congruent, then the lines are parallel" (or corresponding depending on the position).
## Step6: Analyze $ m\angle7 + m\angle19 = 180^\circ $ (Part f)
∠7 and ∠19 are same - side interior angles. By the Converse of Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠7 and ∠19, and the converse is "If same - side interior angles are supplementary, then the lines are parallel".
## Step7: Analyze ∠6 ≅ ∠17 (Part g)
∠6 and ∠17 are corresponding angles. By the Converse of Corresponding Angles Theorem, if corresponding angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal creating ∠6 and ∠17, and the converse is "If corresponding angles are congruent, then the lines are parallel".
## Step8: Analyze ∠9 ≅ ∠24 (Part h)
∠9 and ∠24 are corresponding angles. By the Converse of Corresponding Angles Theorem, if corresponding angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠9 and ∠24, and the converse is "If corresponding angles are congruent, then the lines are parallel".
## Step9: Analyze ∠2 ≅ ∠21 (Part i)
∠2 and ∠21 are corresponding angles. By the Converse of Corresponding Angles Theorem, if corresponding angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal creating ∠2 and ∠21, and the converse is "If corresponding angles are congruent, then the lines are parallel".
## Step10: Analyze $ m\angle3 + m\angle7 = 180^\circ $ (Part j)
∠3 and ∠7 are same - side interior angles. By the Converse of Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠3 and ∠7, and the converse is "If same - side interior angles are supplementary, then the lines are parallel".
## Step11: Analyze ∠6 ≅ ∠11 (Part k)
∠6 and ∠11 are alternate interior angles. By the Converse of Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠6 and ∠11, and the converse is "If alternate interior angles are congruent, then the lines are parallel".
## Step12: Analyze ∠1 ≅ ∠3 (Part l)
∠1 and ∠3 are vertical angles? No, if they are formed by a transversal and two lines, maybe alternate interior or corresponding. Wait, ∠1 and ∠3: if they are alternate interior angles, by the Converse of Alternate Interior Angles Theorem, if congruent, lines are parallel. So the parallel lines are the ones cut by the transversal, and the converse is "If alternate interior angles are congruent, then the lines are parallel" (or corresponding depending on position).
## Step13: Analyze ∠12 ≅ ∠15 (Part m)
∠12 and ∠15 are alternate interior angles. By the Converse of Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠12 and ∠15, and the converse is "If alternate interior angles are congruent, then the lines are parallel".
## Step14: Analyze $ m\angle13 + m\angle16 = 180^\circ $ (Part n)
∠13 and ∠16 are same - side interior angles. By the Converse of Same - Side Interior Angles Theorem, if same - side interior angles are supplementary, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠13 and ∠16, and the converse is "If same - side interior angles are supplementary, then the lines are parallel".
## Step15: Analyze ∠15 ≅ ∠18 (Part o)
∠15 and ∠18 are alternate interior angles. By the Converse of Alternate Interior Angles Theorem, if alternate interior angles are congruent, the lines are parallel. So the parallel lines are the ones cut by the transversal forming ∠15 and ∠18, and the converse is "If alternate interior angles are congruent, then the lines are parallel".

# Answer:
To fill the table, we use the converses of angle - related theorems (Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Same - Side Interior Angles) to determine parallel lines:

| Given | Parallel Lines | Converse |
| --- | --- | --- |
| a. $ \angle10\cong\angle15 $ | Lines cut by transversal (creating $ \angle10,\angle15 $) are parallel | If alternate exterior angles are congruent, then lines are parallel |
| b. $ m\angle14 + m\angle18 = 180^\circ $ | Lines cut by transversal (creating $ \angle14,\angle18 $) are parallel | If same - side interior angles are supplementary, then lines are parallel |
| c. $ \angle4\cong\angle20 $ | Lines cut by transversal (creating $ \angle4,\angle20 $) are parallel | If corresponding angles are congruent, then lines are parallel |
| d. $ \angle3\cong\angle16 $ | Lines cut by transversal (creating $ \angle3,\angle16 $) are parallel | If alternate interior angles are congruent, then lines are parallel |
| e. $ \angle10\cong\angle12 $ | Lines cut by transversal (creating $ \angle10,\angle12 $) are parallel | If alternate interior (or corresponding) angles are congruent, then lines are parallel |
| f. $ m\angle7 + m\angle19 = 180^\circ $ | Lines cut by transversal (creating $ \angle7,\angle19 $) are parallel | If same - side interior angles are supplementary, then lines are parallel |
| g. $ \angle6\cong\angle17 $ | Lines cut by transversal (creating $ \angle6,\angle17 $) are parallel | If corresponding angles are congruent, then lines are parallel |
| h. $ \angle9\cong\angle24 $ | Lines cut by transversal (creating $ \angle9,\angle24 $) are parallel | If corresponding angles are congruent, then lines are parallel |
| i. $ \angle2\cong\angle21 $ | Lines cut by transversal (creating $ \angle2,\angle21 $) are parallel | If corresponding angles are congruent, then lines are parallel |
| j. $ m\angle3 + m\angle7 = 180^\circ $ | Lines cut by transversal (creating $ \angle3,\angle7 $) are parallel | If same - side interior angles are supplementary, then lines are parallel |
| k. $ \angle6\cong\angle11 $ | Lines cut by transversal (creating $ \angle6,\angle11 $) are parallel | If alternate interior angles are congruent, then lines are parallel |
| l. $ \angle1\cong\angle3 $ | Lines cut by transversal (creating $ \angle1,\angle3 $) are parallel | If alternate interior (or corresponding) angles are congruent, then lines are parallel |
| m. $ \angle12\cong\angle15 $ | Lines cut by transversal (creating $ \angle12,\angle15 $) are parallel | If alternate interior angles are congruent, then lines are parallel |
| n. $ m\angle13 + m\angle16 = 180^\circ $ | Lines cut by transversal (creating $ \angle13,\angle16 $) are parallel | If same - side interior angles are supplementary, then lines are parallel |
| o. $ \angle15\cong\angle18 $ | Lines cut by transversal (creating $ \angle15,\angle18 $) are parallel | If alternate interior angles are congruent, then lines are parallel |

Given	Parallel Lines	Converse
b. \( m\angle14 + m\angle18 = 180^\circ \)	Lines cut by transversal (creating \( \angle14,\angle18 \)) are parallel	If same - side interior angles are supplementary, then lines are parallel
c. \( \angle4\cong\angle20 \)	Lines cut by transversal (creating \( \angle4,\angle20 \)) are parallel	If corresponding angles are congruent, then lines are parallel
d. \( \angle3\cong\angle16 \)	Lines cut by transversal (creating \( \angle3,\angle16 \)) are parallel	If alternate interior angles are congruent, then lines are parallel
e. \( \angle10\cong\angle12 \)	Lines cut by transversal (creating \( \angle10,\angle12 \)) are parallel	If alternate interior (or corresponding) angles are congruent, then lines are parallel
f. \( m\angle7 + m\angle19 = 180^\circ \)	Lines cut by transversal (creating \( \angle7,\angle19 \)) are parallel	If same - side interior angles are supplementary, then lines are parallel
g. \( \angle6\cong\angle17 \)	Lines cut by transversal (creating \( \angle6,\angle17 \)) are parallel	If corresponding angles are congruent, then lines are parallel
h. \( \angle9\cong\angle24 \)	Lines cut by transversal (creating \( \angle9,\angle24 \)) are parallel	If corresponding angles are congruent, then lines are parallel
i. \( \angle2\cong\angle21 \)	Lines cut by transversal (creating \( \angle2,\angle21 \)) are parallel	If corresponding angles are congruent, then lines are parallel
j. \( m\angle3 + m\angle7 = 180^\circ \)	Lines cut by transversal (creating \( \angle3,\angle7 \)) are parallel	If same - side interior angles are supplementary, then lines are parallel
k. \( \angle6\cong\angle11 \)	Lines cut by transversal (creating \( \angle6,\angle11 \)) are parallel	If alternate interior angles are congruent, then lines are parallel
l. \( \angle1\cong\angle3 \)	Lines cut by transversal (creating \( \angle1,\angle3 \)) are parallel	If alternate interior (or corresponding) angles are congruent, then lines are parallel
m. \( \angle12\cong\angle15 \)	Lines cut by transversal (creating \( \angle12,\angle15 \)) are parallel	If alternate interior angles are congruent, then lines are parallel
n. \( m\angle13 + m\angle16 = 180^\circ \)	Lines cut by transversal (creating \( \angle13,\angle16 \)) are parallel	If same - side interior angles are supplementary, then lines are parallel
o. \( \angle15\cong\angle18 \)	Lines cut by transversal (creating \( \angle15,\angle18 \)) are parallel	If alternate interior angles are congruent, then lines are parallel

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze ∠10 ≅ ∠15 (Part a)

Step2: Analyze \( m\angle14 + m\angle18 = 180^\circ \) (Part b)

Step3: Analyze ∠4 ≅ ∠20 (Part c)

Step4: Analyze ∠3 ≅ ∠16 (Part d)

Step5: Analyze ∠10 ≅ ∠12 (Part e)

Step6: Analyze \( m\angle7 + m\angle19 = 180^\circ \) (Part f)

Step7: Analyze ∠6 ≅ ∠17 (Part g)

Step8: Analyze ∠9 ≅ ∠24 (Part h)

Step9: Analyze ∠2 ≅ ∠21 (Part i)

Answer: