Question

use the given information to decide if ( aparallel b,mparallel l ), or neither. if they are parallel, state the reason.
a. ( mangle1 = mangle3 )
b. ( mangle3 = mangle6 )
c. ( mangle9 = mangle12 )
d. ( angle6 ) and ( angle10 ) are supplementary
e. ( mangle2 = mangle11 )
f. ( mangle11 ) and ( angle9 ) are supplementary
g. ( mangle2 ) and ( angle3 ) are supplementary
h. ( mangle6 = mangle7 )

Explanation:

Step1: Analyze part a

$\angle1$ and $\angle3$ are corresponding angles. If corresponding angles are equal, then the lines are parallel. Here, $m\angle1 = m\angle3$, so $a\parallel b$ (Corresponding Angles Postulate).

Step2: Analyze part b

$\angle3$ and $\angle6$ are alternate - interior angles. If alternate - interior angles are equal, then the lines are parallel. Since $m\angle3=m\angle6$, so $a\parallel b$ (Alternate Interior Angles Theorem).

Step3: Analyze part c

$\angle9$ and $\angle12$ are vertical angles. Vertical angles are always equal, but this does not imply that any of the lines ($a,b,l,m$) are parallel.

Step4: Analyze part d

$\angle6$ and $\angle10$ are same - side interior angles. If same - side interior angles are supplementary, then the lines are parallel. Since $\angle6$ and $\angle10$ are supplementary, so $l\parallel m$ (Same - Side Interior Angles Theorem).

Step5: Analyze part e

$\angle2$ and $\angle11$ are not in a position (corresponding, alternate - interior, same - side interior) to imply parallelism of any of the lines $a,b,l,m$.

Step6: Analyze part f

$\angle11$ and $\angle9$ are adjacent angles. Their supplementary relationship does not imply parallelism of any of the lines $a,b,l,m$.

Step7: Analyze part g

$\angle2$ and $\angle3$ are same - side interior angles. If same - side interior angles are supplementary, then the lines are parallel. Since $m\angle2$ and $m\angle3$ are supplementary, so $l\parallel m$ (Same - Side Interior Angles Theorem).

Step8: Analyze part h

$\angle6$ and $\angle7$ are alternate - interior angles. If alternate - interior angles are equal, then the lines are parallel. Since $m\angle6 = m\angle7$, so $l\parallel m$ (Alternate Interior Angles Theorem).

Answer:

a. $a\parallel b$, Corresponding Angles Postulate
b. $a\parallel b$, Alternate Interior Angles Theorem
c. Neither
d. $l\parallel m$, Same - Side Interior Angles Theorem
e. Neither
f. Neither
g. $l\parallel m$, Same - Side Interior Angles Theorem
h. $l\parallel m$, Alternate Interior Angles Theorem