Question

1. if ( mangle 9 = 97^{circ} ) and ( mangle 12 = 114^{circ} ), find each measure. a. ( mangle 1 = 97 ) b. ( mangle 2 = 83 ) c. ( mangle 3 = 63 ) d. ( mangle 4 = 97 ) e. ( mangle 5 = 83 ) f. ( mangle 6 = 97 ) g. ( mangle 7 = ) h. ( mangle 8 = ) l. ( mangle 10 = 83 ) j. ( mangle 11 = 66 ) k. ( mangle 13 = ) l. ( mangle 14 = 97 ) m. ( mangle 15 = 114 ) n. ( mangle 16 = 114 )

Explanation:

Step1: Vertical Angles

Vertical angles are equal. Since \(m\angle9 = 97^{\circ}\), and \(\angle1\) and \(\angle9\) are vertical angles, so \(m\angle1=97^{\circ}\) (a).

Step2: Linear Pair

\(\angle2\) and \(\angle9\) form a linear pair (\(\angle2+\angle9 = 180^{\circ}\)). Then \(m\angle2=180 - 97=83^{\circ}\) (b).

Step3: Vertical Angles

\(\angle3\) and \(\angle2\) are vertical angles. So \(m\angle3 = m\angle2=83^{\circ}\) (c).

Step4: Vertical Angles

\(\angle4\) and \(\angle9\) are vertical angles. So \(m\angle4 = m\angle9=97^{\circ}\) (d).

Step5: Linear Pair

\(\angle5\) and \(\angle12\) form a linear pair (\(\angle5+\angle12 = 180^{\circ}\)). Given \(m\angle12 = 114^{\circ}\), then \(m\angle5=180 - 114 = 66^{\circ}\) (e).

Step6: Vertical Angles

\(\angle6\) and \(\angle9\) are vertical angles. So \(m\angle6 = m\angle9=97^{\circ}\) (f).

Step7: Linear Pair

\(\angle7\) and \(\angle12\) form a linear pair (\(\angle7+\angle12 = 180^{\circ}\)). Then \(m\angle7=180 - 114 = 66^{\circ}\) (g).

Step8: Given

\(m\angle8 = 70^{\circ}\) (h).

Step9: Vertical Angles

\(\angle10\) and \(\angle2\) are vertical angles. So \(m\angle10 = m\angle2=83^{\circ}\) (i).

Step10: Linear Pair

\(\angle11\) and \(\angle12\) form a linear pair (\(\angle11+\angle12 = 180^{\circ}\)). Then \(m\angle11=180 - 114 = 66^{\circ}\) (j).

Step11: Vertical Angles

\(\angle13\) and \(\angle9\) are vertical angles. So \(m\angle13 = m\angle9=97^{\circ}\) (k).

Step12: Vertical Angles

\(\angle14\) and \(\angle9\) are vertical angles. So \(m\angle14 = m\angle9=97^{\circ}\) (l).

Step13: Vertical Angles

\(\angle15\) and \(\angle12\) are vertical angles. So \(m\angle15 = m\angle12=114^{\circ}\) (m).

Step14: Vertical Angles

\(\angle16\) and \(\angle12\) are vertical angles. So \(m\angle16 = m\angle12=114^{\circ}\) (n).

Answer:

a. \(m\angle1 = 97^{\circ}\)
b. \(m\angle2 = 83^{\circ}\)
c. \(m\angle3 = 83^{\circ}\)
d. \(m\angle4 = 97^{\circ}\)
e. \(m\angle5 = 66^{\circ}\)
f. \(m\angle6 = 97^{\circ}\)
g. \(m\angle7 = 66^{\circ}\)
h. \(m\angle8 = 70^{\circ}\)
i. \(m\angle10 = 83^{\circ}\)
j. \(m\angle11 = 66^{\circ}\)
k. \(m\angle13 = 97^{\circ}\)
l. \(m\angle14 = 97^{\circ}\)
m. \(m\angle15 = 114^{\circ}\)
n. \(m\angle16 = 114^{\circ}\)