Question

unit 3: parallel & perpendicular lines
homework 3: proving lines are parallel
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determine if l || m based on the information given on the diagram. if yes, state the converse that proves the lines are parallel.

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

Step2: Analyze each given condition

a. $\angle10\cong\angle15$

These are alternate interior angles. If $\angle10\cong\angle15$, then $l\parallel m$ by the Alternate Interior Angles Converse.

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

These are same - side interior angles. If $m\angle14 + m\angle18 = 180^{\circ}$, then $l\parallel m$ by the Same - Side Interior Angles Converse.

c. $\angle4\cong\angle20$

These are corresponding angles. If $\angle4\cong\angle20$, then $l\parallel m$ by the Corresponding Angles Converse.

d. $\angle3\cong\angle16$

These are alternate exterior angles. If $\angle3\cong\angle16$, then $l\parallel m$ by the Alternate Exterior Angles Converse.

e. $\angle10\cong\angle12$

These angles do not have a parallel - line relationship. There is no parallelism conclusion.

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

These are same - side interior angles. If $m\angle7 + m\angle19 = 180^{\circ}$, then $l\parallel m$ by the Same - Side Interior Angles Converse.

g. $\angle6\cong\angle17$

These are corresponding angles. If $\angle6\cong\angle17$, then $l\parallel m$ by the Corresponding Angles Converse.

h. $\angle9\cong\angle24$

These are corresponding angles. If $\angle9\cong\angle24$, then $l\parallel m$ by the Corresponding Angles Converse.

i. $\angle2\cong\angle21$

These are corresponding angles. If $\angle2\cong\angle21$, then $l\parallel m$ by the Corresponding Angles Converse.

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

These are same - side interior angles. If $m\angle3 + m\angle7 = 180^{\circ}$, then $l\parallel m$ by the Same - Side Interior Angles Converse.

k. $\angle6\cong\angle11$

These angles do not have a parallel - line relationship. There is no parallelism conclusion.

l. $\angle1\cong\angle3$

These are vertical angles and do not imply parallel lines. There is no parallelism conclusion.

m. $\angle12\cong\angle15$

These angles do not have a parallel - line relationship. There is no parallelism conclusion.

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

These are same - side interior angles. If $m\angle13 + m\angle16 = 180^{\circ}$, then $l\parallel m$ by the Same - Side Interior Angles Converse.

o. $\angle15\cong\angle18$

These angles do not have a parallel - line relationship. There is no parallelism conclusion.

Answer:

a. Parallel Lines: $l\parallel m$, Converse: Alternate Interior Angles Converse
b. Parallel Lines: $l\parallel m$, Converse: Same - Side Interior Angles Converse
c. Parallel Lines: $l\parallel m$, Converse: Corresponding Angles Converse
d. Parallel Lines: $l\parallel m$, Converse: Alternate Exterior Angles Converse
e. No parallelism conclusion
f. Parallel Lines: $l\parallel m$, Converse: Same - Side Interior Angles Converse
g. Parallel Lines: $l\parallel m$, Converse: Corresponding Angles Converse
h. Parallel Lines: $l\parallel m$, Converse: Corresponding Angles Converse
i. Parallel Lines: $l\parallel m$, Converse: Corresponding Angles Converse
j. Parallel Lines: $l\parallel m$, Converse: Same - Side Interior Angles Converse
k. No parallelism conclusion
l. No parallelism conclusion
m. No parallelism conclusion
n. Parallel Lines: $l\parallel m$, Converse: Same - Side Interior Angles Converse
o. No parallelism conclusion