Question

example 2 given ( mangle6 = 142^circ ), find the measure of each missing angle. give your reasoning.
a. ( mangle1=)
b. ( mangle2=)
c. ( mangle3=)
d. ( mangle4=)
e. ( mangle5=)
f. ( mangle7=)
g. ( mangle8=)

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. So \(m\angle1 = m\angle2\), \(m\angle3 = m\angle4\), \(m\angle5 = m\angle6\), \(m\angle7 = m\angle8\)

Step2: Use the property of supplementary angles

\(\angle2\) and \(\angle6\) are supplementary. So \(m\angle2=180^{\circ}-m\angle6 = 180^{\circ}- 142^{\circ}=38^{\circ}\). Then \(m\angle1 = 38^{\circ}\)

Step3: Use the property of vertical angles and parallel - line properties (assuming the lines are parallel)

Since \(\angle2\) and \(\angle3\) are vertical angles, \(m\angle3 = 38^{\circ}\). And \(m\angle4=m\angle3 = 38^{\circ}\)
Since \(\angle5\) and \(\angle6\) are vertical angles, \(m\angle5 = 142^{\circ}\)
Since \(\angle6\) and \(\angle8\) are supplementary (\(m\angle8=180 - 142=38^{\circ}\)) and \(\angle7\) and \(\angle8\) are vertical angles, \(m\angle7 = 38^{\circ}\)

Answer:

a. \(m\angle1 = 38^{\circ}\) (vertical angles with \(\angle2\), and \(\angle2\) is supplementary to \(\angle6\))
b. \(m\angle2 = 38^{\circ}\) (\(\angle2\) and \(\angle6\) are supplementary: \(180 - 142\))
c. \(m\angle3 = 38^{\circ}\) (vertical angles with \(\angle2\))
d. \(m\angle4 = 38^{\circ}\) (vertical angles with \(\angle3\))
e. \(m\angle5 = 142^{\circ}\) (vertical angles with \(\angle6\))
f. \(m\angle7 = 38^{\circ}\) (vertical angles with \(\angle8\), and \(\angle8\) is supplementary to \(\angle6\))
g. \(m\angle8 = 38^{\circ}\) (\(\angle8\) and \(\angle6\) are supplementary: \(180 - 142\))