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: Find \(m\angle2\)

\(\angle2\) and \(\angle6\) are vertical angles. Vertical angles are equal. So \(m\angle2 = m\angle6=142^{\circ}\)

Step2: Find \(m\angle1\)

\(\angle1\) and \(\angle2\) are supplementary (they form a linear - pair). So \(m\angle1=180^{\circ}-m\angle2\). Substitute \(m\angle2 = 142^{\circ}\), then \(m\angle1=180 - 142=38^{\circ}\)

Step3: Find \(m\angle3\)

\(\angle1\) and \(\angle3\) are vertical angles. Vertical angles are equal. So \(m\angle3=m\angle1 = 38^{\circ}\)

Step4: Find \(m\angle4\)

\(\angle2\) and \(\angle4\) are vertical angles. Vertical angles are equal. So \(m\angle4=m\angle2 = 142^{\circ}\)

Step5: Find \(m\angle5\)

\(\angle1\) and \(\angle5\) are vertical angles. Vertical angles are equal. So \(m\angle5=m\angle1 = 38^{\circ}\)

Step6: Find \(m\angle7\)

\(\angle6\) and \(\angle7\) are supplementary (they form a linear - pair). So \(m\angle7=180^{\circ}-m\angle6\). Substitute \(m\angle6 = 142^{\circ}\), then \(m\angle7=180 - 142=38^{\circ}\)

Step7: Find \(m\angle8\)

\(\angle6\) and \(\angle8\) are vertical angles. Vertical angles are equal. So \(m\angle8=m\angle6 = 142^{\circ}\)

Answer:

a. \(m\angle1 = 38^{\circ}\) (Supplementary to \(\angle2\), \(\angle1+\angle2 = 180^{\circ}\))
b. \(m\angle2 = 142^{\circ}\) (Vertical angles with \(\angle6\))
c. \(m\angle3 = 38^{\circ}\) (Vertical angles with \(\angle1\))
d. \(m\angle4 = 142^{\circ}\) (Vertical angles with \(\angle2\))
e. \(m\angle5 = 38^{\circ}\) (Vertical angles with \(\angle1\))
f. \(m\angle7 = 38^{\circ}\) (Supplementary to \(\angle6\), \(\angle6+\angle7 = 180^{\circ}\))
g. \(m\angle8 = 142^{\circ}\) (Vertical angles with \(\angle6\))