Question

example 3 given ( mangle5 = 82^{circ} ), find the measure of each missing angle. a. ( mangle1 = ) e. ( mangle6 = ) i. ( mangle10 = ) b. ( mangle2 = ) f. ( mangle7 = ) j. ( mangle11 = ) c. ( mangle3 = ) g. ( mangle8 = ) k. ( mangle12 = ) d. ( mangle4 = ) h. ( mangle9 = )

Explanation:

Step1: Vertical angles

Vertical angles are equal. So \(m\angle5 = m\angle11=82^{\circ}\) (vertical angles). Also, \(m\angle5 + m\angle6=180^{\circ}\) (linear - pair of angles). Then \(m\angle6 = 180^{\circ}-82^{\circ}=98^{\circ}\). Since \(m\angle6 = m\angle12 = 98^{\circ}\) (vertical angles) and \(m\angle5=m\angle9 = 82^{\circ}\) (vertical angles).

Step2: Corresponding and alternate - interior angles

Corresponding angles: \(\angle5\) and \(\angle10\) are corresponding angles. So \(m\angle10 = 82^{\circ}\). Alternate - interior angles: \(\angle1\) and \(\angle5\) are not in a direct parallel - line relationship. But \(\angle1\) and \(\angle4\) are vertical angles, \(\angle2\) and \(\angle5\) are not in a direct parallel - line relationship. However, if we assume the lines are parallel (implied by the angle - finding context).
For parallel lines cut by a transversal:

• \(m\angle1=m\angle5 = 82^{\circ}\) (corresponding angles if we consider the first and second set of lines in a parallel - transversal setup).
• \(m\angle2=m\angle6 = 98^{\circ}\) (corresponding angles).
• \(m\angle3=m\angle6 = 98^{\circ}\) (corresponding angles).
• \(m\angle4=m\angle1 = 82^{\circ}\) (vertical angles).
• \(m\angle7=m\angle1 = 82^{\circ}\) (vertical angles).
• \(m\angle8=m\angle2 = 98^{\circ}\) (vertical angles).

Answer:

a. \(m\angle1 = 82^{\circ}\)
b. \(m\angle2 = 98^{\circ}\)
c. \(m\angle3 = 98^{\circ}\)
d. \(m\angle4 = 82^{\circ}\)
e. \(m\angle6 = 98^{\circ}\)
f. \(m\angle7 = 82^{\circ}\)
g. \(m\angle8 = 98^{\circ}\)
h. \(m\angle9 = 82^{\circ}\)
i. \(m\angle10 = 82^{\circ}\)
j. \(m\angle11 = 82^{\circ}\)
k. \(m\angle12 = 98^{\circ}\)