Question

ex2: use the given information to decide if ( a||b,m||l ), or neither. if they are parallel, state the reason.
a. ( m<1 = m<3 )
b. ( m<3 = m<6 )
c. ( m<9 = m<12 )
d. ( <6 ) and ( <10 ) are supplementary
e. ( m<2 = m<11 )
f. ( m<11 ) and ( <9 ) are supplementary
g. ( m<2 ) and ( <3 ) are supplementary
h. ( m<6 = m<7 )

Explanation:

Step1: Analyze part a

$\angle1$ and $\angle3$ are corresponding angles. If $m\angle1 = m\angle3$, then by the Corresponding Angles Theorem, $a\parallel b$.

Step2: Analyze part b

$\angle3$ and $\angle6$ are alternate - interior angles. If $m\angle3 = m\angle6$, then by the Alternate - Interior Angles Theorem, $a\parallel b$.

Step3: Analyze part c

$\angle9$ and $\angle12$ are vertical angles. Vertical angles are always equal, but this does not imply that any lines are parallel.

Step4: Analyze part d

$\angle6$ and $\angle10$ are same - side interior angles. If they are supplementary, then by the Same - Side Interior Angles Theorem, $a\parallel b$.

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 and supplementary (they form a linear pair). This is a property of angles on a straight line and does not imply parallelism.

Step7: Analyze part g

$\angle2$ and $\angle3$ are same - side interior angles. If they are supplementary, then by the Same - Side Interior Angles Theorem, $l\parallel m$.

Step8: Analyze part h

$\angle6$ and $\angle7$ are alternate - exterior angles. If $m\angle6 = m\angle7$, then by the Alternate - Exterior Angles Theorem, $l\parallel m$.

Answer:

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