Question

in the figure below, ( lr^{leftrightarrow} ), ( ms^{leftrightarrow} ), and ( nt^{leftrightarrow} ) all intersect at point ( a ), with angle measures as marked. what are the measures of the angles listed below? (5 pts - 1 each)

• provide all theorems / reasoning and show all work**

a. ( mangle lam=)
b. ( mangle nat=)
c. ( mangle tal=)
d. ( mangle tar=)
e. ( mangle las=)

Explanation:

Step1: Find \(m\angle LAM\)

Vertical angles are equal. \(\angle LAM\) and the \(68^{\circ}\) angle are vertical angles. So \(m\angle LAM = 68^{\circ}\)

Step2: Find \(m\angle NAT\)

Vertical angles are equal. \(\angle NAT\) and the \(68^{\circ}\) angle are not related directly. But using the fact that the sum of angles around a point \(A\) for a straight - line formed by \(LN\) and \(TR\) (assuming the lines form a complete set of angles around \(A\)). Wait, no, more accurately, for the intersection of lines:
We know that \(\angle MAN = 42^{\circ}\), \(\angle SAR=68^{\circ}\). Since \(\angle NAT\) and \(\angle MAS\) are vertical angles. And using the fact that the sum of angles around a point \(A\) for the intersection of \(MS\) and \(LR\) and \(NT\):
\(\angle NAT\) and \(\angle MAS\) are vertical angles. But another way: \(\angle NAT\) and the angle adjacent to \(42^{\circ}\) and \(68^{\circ}\) (using the straight - line property \(180^{\circ}\)). Wait, no.
Since \(\angle LAM = 68^{\circ}\), \(\angle MAN=42^{\circ}\), and \(\angle NAT\) forms a straight - line with \(\angle LAM\) and \(\angle MAN\) (no, wrong). Wait, correct approach:
\(\angle NAT\) and \(\angle MAS\) are vertical angles. But using the fact that \(\angle LAM = 68^{\circ}\), \(\angle MAN = 42^{\circ}\), and for the line \(LN\) (assuming \(LN\) is a straight - line, but no, the lines \(LR\), \(MS\), \(NT\) intersect at \(A\)).
Wait, using the vertical - angle theorem: \(\angle NAT\) and \(\angle MAS\) are vertical angles. But also, using the fact that \(\angle LAM = 68^{\circ}\), \(\angle MAN=42^{\circ}\), and \(\angle NAT = 180^{\circ}-\angle LAM-\angle MAN\) (if \(LN\) is a straight - line, but no, the lines intersect at \(A\)). Wait, no, correct:
\(\angle NAT\) and \(\angle MAS\) are vertical angles. But \(\angle MAS\) can be found using the fact that \(\angle LAM = 68^{\circ}\), \(\angle MAN = 42^{\circ}\), and \(\angle MAS=180^{\circ}-\angle LAM - \angle MAN\) (if we consider the angles around point \(A\) for a certain combination. Wait, no, correct:
Since \(\angle LAM = 68^{\circ}\), \(\angle MAN = 42^{\circ}\), and \(\angle NAT\) is supplementary to \(\angle LAM\) and \(\angle MAN\) (if we assume that \(LN\) is a straight - line, but actually, using the vertical - angle and angle - addition postulate.
Wait, another approach:
We know that \(\angle LAM\) and \(\angle SAR\) are vertical angles (\(m\angle LAM=m\angle SAR = 68^{\circ}\)), \(\angle MAN\) and \(\angle TAS\) are vertical angles (\(m\angle MAN=m\angle TAS = 42^{\circ}\))
\(m\angle NAT=180^{\circ}-m\angle MAN - m\angle LAM\) (because \(\angle NAT+\angle MAN+\angle LAM = 180^{\circ}\) if we consider the straight - line \(LN\) (but actually, it's the sum of angles around the intersection point for a particular linear - pair - like combination).
\(m\angle NAT=180^{\circ}-42^{\circ}-68^{\circ}=70^{\circ}\)

Step3: Find \(m\angle TAL\)

\(\angle TAL\) and \(\angle RAN\) are vertical angles. \(\angle RAN=\angle MAN+\angle LAM\) (angle - addition postulate). But also, using the fact that \(\angle TAL\) and \(\angle RAN\) are vertical angles.
Another way: \(\angle TAL = 180^{\circ}-\angle NAT\) (if \(NT\) is a straight - line). \(m\angle TAL = 180^{\circ}-70^{\circ}=110^{\circ}\)

Step4: Find \(m\angle TAR\)

\(\angle TAR\) and \(\angle TAL\) are supplementary (if \(LR\) is a straight - line). \(m\angle TAR=180^{\circ}-m\angle TAL\)
Since \(m\angle TAL = 110^{\circ}\), \(m\angle TAR = 70^{\circ}\)

Step5: Find \(m\angle LAS\)

\(\angle LAS=\angle LAM+\angle MAS\)
We know \(m\angle LAM = 68^{\circ}\), \(m\angle MAS=m\angle NAT = 70^{\circ}\) (vertical angles)
\(m\angle LAS=68^{\circ}+70^{\circ}=138^{\circ}\)

Answer:

a. \(68^{\circ}\)
b. \(70^{\circ}\)
c. \(110^{\circ}\)
d. \(70^{\circ}\)
e. \(138^{\circ}\)