Question

practice
in the figure, ( m angle 9 = 80 ) and ( m angle 5 = 68 ). find the measure
of each angle. tell which postulate(s) or theorem(s) you used.
a. ( angle 12 )
b. ( angle 1 )
c. ( angle 4 )
d. ( angle 3 )
e. ( angle 7 )
f. ( angle 16 )

Explanation:

Step1: Find \(m\angle12\)

\(\angle9\) and \(\angle12\) are vertical angles. Vertical angles are congruent. So \(m\angle12 = m\angle9\). Given \(m\angle9 = 80^{\circ}\), so \(m\angle12=80^{\circ}\) (Vertical Angles Theorem).

Step2: Find \(m\angle1\)

\(\angle5\) and \(\angle1\) are corresponding angles. Corresponding angles are congruent when lines are parallel. Given \(m\angle5 = 68^{\circ}\), so \(m\angle1 = 68^{\circ}\) (Corresponding Angles Postulate).

Step3: Find \(m\angle4\)

\(\angle1\) and \(\angle4\) are supplementary (they form a linear - pair). The sum of angles in a linear - pair is \(180^{\circ}\). So \(m\angle4=180 - m\angle1\). Substitute \(m\angle1 = 68^{\circ}\), then \(m\angle4=180 - 68=112^{\circ}\) (Linear - Pair Postulate).

Step4: Find \(m\angle3\)

\(\angle4\) and \(\angle3\) are vertical angles. Vertical angles are congruent. So \(m\angle3 = m\angle4\). Since \(m\angle4 = 112^{\circ}\), then \(m\angle3=112^{\circ}\) (Vertical Angles Theorem).

Step5: Find \(m\angle7\)

\(\angle5\) and \(\angle13\) are corresponding angles (\(l_1\parallel l_2\), transversal \(t\)). \(m\angle13 = m\angle5 = 68^{\circ}\) (Corresponding Angles Postulate). \(\angle13\) and \(\angle7\) are alternate - exterior angles (\(l_3\parallel l_4\), transversal \(t\)). So \(m\angle7 = m\angle13=68^{\circ}\) (Alternate - Exterior Angles Theorem).

Step6: Find \(m\angle16\)

\(\angle12\) and \(\angle16\) are corresponding angles (\(l_3\parallel l_4\), transversal \(s\)). So \(m\angle16 = m\angle12\). Since \(m\angle12 = 80^{\circ}\), then \(m\angle16 = 80^{\circ}\) (Corresponding Angles Postulate).

Answer:

a. \(m\angle12 = 80^{\circ}\) (Vertical Angles Theorem).
b. \(m\angle1 = 68^{\circ}\) (Corresponding Angles Postulate).
c. \(m\angle4 = 112^{\circ}\) (Linear - Pair Postulate).
d. \(m\angle3 = 112^{\circ}\) (Vertical Angles Theorem).
e. \(m\angle7 = 68^{\circ}\) (Corresponding Angles Postulate and Alternate - Exterior Angles Theorem).
f. \(m\angle16 = 80^{\circ}\) (Corresponding Angles Postulate).