Question

1. if ( mangle9 = 97^{circ} ) and ( mangle12 = 114^{circ} ), find each measure.

a. ( mangle1 = 97 )
b. ( mangle2 = 83 )
c. ( mangle3 = )
d. ( mangle4 = )
e. ( mangle5 = 83 )
f. ( mangle6 = 97 )
g. ( mangle7 = )
h. ( mangle8 = )
i. ( mangle10 = 83 )
j. ( mangle11 = 66 )
k. ( mangle13 = )
l. ( mangle14 = 97 )
m. ( mangle15 = 114 )
n. ( mangle16 = 114 )

Explanation:

Step1: Find \(m\angle3\)

\(\angle2\) and \(\angle3\) are supplementary with \(\angle12\). Since \(m\angle12 = 114^{\circ}\), and \(\angle2+\angle3+\angle12=180^{\circ}\), but also \(\angle2 = 83^{\circ}\) (given in the problem's partial answers). Using the fact that \(\angle2\) and \(\angle3\) are vertical - angle - related in a way (alternate interior angles or linear - pair - adjusted). Wait, another approach: \(\angle3\) and \(\angle12\) and \(\angle2\) form a triangle - like angle sum. But more simply, \(\angle3=180^{\circ}-m\angle2 - m\angle12\) is wrong. Wait, no, actually \(\angle3\) and \(\angle12\) are not directly in a triangle. Wait, looking at the parallel - line and transversal properties. \(\angle3\) and \(\angle5\) are vertical angles. Wait, no. Wait, \(\angle3\): \(\angle1 = 97^{\circ}\), \(\angle2=83^{\circ}\), \(\angle3\) and \(\angle2\) are in a linear pair with \(\angle12\). No, better: \(\angle3\) and \(\angle5\) are vertical angles. Since \(\angle5 = 83^{\circ}\) (given as \(m\angle5 = 83^{\circ}\)), so \(m\angle3=83^{\circ}\) (vertical angles are equal).

Step2: Find \(m\angle4\)

\(\angle3\) and \(\angle4\) are supplementary (linear pair). So \(m\angle4=180^{\circ}-m\angle3\). Substitute \(m\angle3 = 83^{\circ}\), then \(m\angle4=180 - 83=97^{\circ}\).

Answer:

c. \(m\angle3 = 83^{\circ}\), d. \(m\angle4=97^{\circ}\)