Question

example 4 given ( mangle12 = 121^{circ} ) and ( mangle6 = 75^{circ} ), find the measure of each missing angle

Explanation:

Step1: Use linear - pair relationship

Angles that form a linear pair are supplementary (sum to \(180^{\circ}\)).
For \(\angle1\) and \(\angle2\): \(\angle1+\angle2 = 180^{\circ}\), \(\angle2+\angle3 = 180^{\circ}\), \(\angle4+\angle5 = 180^{\circ}\), \(\angle5+\angle6 = 180^{\circ}\), \(\angle7+\angle8 = 180^{\circ}\), \(\angle8+\angle12 = 180^{\circ}\), \(\angle9+\angle10 = 180^{\circ}\), \(\angle10+\angle14 = 180^{\circ}\), \(\angle11+\angle12 = 180^{\circ}\), \(\angle12+\angle13 = 180^{\circ}\).
Also, vertical angles are equal (\(\angle1=\angle3\), \(\angle2=\angle4\), \(\angle5=\angle7\), \(\angle6=\angle8\), \(\angle9=\angle11\), \(\angle10=\angle14\), \(\angle12=\angle13\)).
Since \(\angle6 = 75^{\circ}\), then \(\angle5=180 - 75=105^{\circ}\) (linear - pair with \(\angle6\)). \(\angle4=\angle2\) (vertical angles), \(\angle1=\angle3\) (vertical angles).
Since \(\angle12 = 121^{\circ}\), then \(\angle11 = 180 - 121=59^{\circ}\) (linear - pair with \(\angle12\)), \(\angle8=\angle6 = 75^{\circ}\) (vertical angles), \(\angle7 = 180 - 75=105^{\circ}\) (linear - pair with \(\angle8\)), \(\angle9=\angle11 = 59^{\circ}\) (vertical angles), \(\angle10=180 - 59 = 121^{\circ}\) (linear - pair with \(\angle9\)), \(\angle13=\angle12 = 121^{\circ}\) (vertical angles), \(\angle14=\angle10 = 121^{\circ}\) (vertical angles).
For \(\angle2\) and \(\angle12\): \(\angle2\) and \(\angle12\) are corresponding angles. But also, \(\angle1\) and \(\angle11\) are corresponding angles.
Since \(\angle11 = 59^{\circ}\), then \(\angle1 = 59^{\circ}\) (corresponding angles), \(\angle2=180 - 59=121^{\circ}\) (linear - pair with \(\angle1\)), \(\angle3=\angle1 = 59^{\circ}\), \(\angle4=\angle2 = 121^{\circ}\)

Answer:

a. \(m\angle1 = 59^{\circ}\)
b. \(m\angle2 = 121^{\circ}\)
c. \(m\angle3 = 59^{\circ}\)
d. \(m\angle4 = 121^{\circ}\)
e. \(m\angle5 = 105^{\circ}\)
f. \(m\angle7 = 105^{\circ}\)
g. \(m\angle8 = 75^{\circ}\)
h. \(m\angle9 = 59^{\circ}\)
i. \(m\angle10 = 121^{\circ}\)
j. \(m\angle11 = 59^{\circ}\)
k. \(m\angle13 = 121^{\circ}\)
l. \(m\angle14 = 121^{\circ}\)