Question

topic 2: parallel lines & angles

1. if ( p parallel q ), ( m angle 7 = 131 ^ { circ } ), and ( m angle 16 = 88 ^ { circ } ), find the measure of each missing angle.

12 if ( a parallel b ), ( m angle 2 = 63 ^ { circ } ), and ( m angle 9 = 105 ^ { circ } ), find the measure of each missing angle.

Explanation:

Step1: Use vertical - angle and linear - pair relationships

Vertical angles are equal. For example, \(\angle1\) and \(\angle7\) are vertical angles. So \(m\angle1 = m\angle7=131^{\circ}\).
A linear pair of angles sums to \(180^{\circ}\). Since \(m\angle7 = 131^{\circ}\), and \(\angle2\) and \(\angle7\) form a linear pair, then \(m\angle2=180 - 131=49^{\circ}\).
\(\angle3\) and \(\angle2\) are vertical angles, so \(m\angle3 = m\angle2 = 49^{\circ}\).
\(\angle4\) and \(\angle1\) are vertical angles, so \(m\angle4 = m\angle1=131^{\circ}\).
\(\angle5\) and \(\angle3\) are vertical angles, so \(m\angle5 = m\angle3 = 49^{\circ}\).
\(\angle6\) and \(\angle2\) are vertical angles, so \(m\angle6 = m\angle2 = 49^{\circ}\).
\(\angle8\) and \(\angle7\) are vertical angles, so \(m\angle8 = m\angle7=131^{\circ}\).

Step2: Use parallel - line (corresponding, alternate - interior, alternate - exterior, consecutive - interior) and angle - sum relationships

Since \(m\angle16 = 88^{\circ}\), and \(\angle9\) and \(\angle16\) are vertical angles, \(m\angle9 = 88^{\circ}\).
\(\angle10\) and \(\angle9\) form a linear pair (\(m\angle10=180 - 88 = 92^{\circ}\)).
\(\angle15\) and \(\angle9\) are vertical angles (\(m\angle15 = 88^{\circ}\)).
\(\angle14\) and \(\angle16\) are vertical angles (\(m\angle14 = 88^{\circ}\)).
For \(\angle11\): \(\angle11\) and \(\angle16\) are alternate - exterior angles (because \(p\parallel q\)), so \(m\angle11 = 88^{\circ}\).
For \(\angle12\): \(\angle12\) and \(\angle11\) form a linear pair (\(m\angle12=180 - 88 = 92^{\circ}\)).
For \(\angle13\): \(\angle13\) and \(\angle11\) are vertical angles (\(m\angle13 = 88^{\circ}\)).

Answer:

a. \(m\angle1 = 131^{\circ}\)
b. \(m\angle2 = 49^{\circ}\)
c. \(m\angle3 = 49^{\circ}\)
d. \(m\angle4 = 131^{\circ}\)
e. \(m\angle5 = 49^{\circ}\)
f. \(m\angle6 = 49^{\circ}\)
g. \(m\angle8 = 131^{\circ}\)
h. \(m\angle9 = 88^{\circ}\)
i. \(m\angle10 = 92^{\circ}\)
j. \(m\angle11 = 88^{\circ}\)
k. \(m\angle12 = 92^{\circ}\)
l. \(m\angle13 = 88^{\circ}\)
m. \(m\angle14 = 88^{\circ}\)
n. \(m\angle15 = 88^{\circ}\)