Question

1. if ( mangle 3 = 54^circ ), find the measure of each missing angle.

Explanation:

Step1: Vertical angles and right - angle

Since \(m\angle2 = 90^{\circ}\) (right - angle).
For \(m\angle1\), \(\angle1\) and \(\angle3\) are complementary (because \(\angle1+\angle2+\angle3 = 180^{\circ}\) and \(\angle2 = 90^{\circ}\)), so \(m\angle1=90^{\circ}-m\angle3\). Given \(m\angle3 = 54^{\circ}\), then \(m\angle1=90 - 54=36^{\circ}\).

Step2: Vertical angles

\(\angle3\) and \(\angle4\) are vertical angles. By the vertical - angles theorem (\(\angle3\cong\angle4\)), so \(m\angle4 = 54^{\circ}\).
\(\angle3\) and \(\angle5\) are supplementary (\(\angle3+\angle5 = 180^{\circ}\)), so \(m\angle5=180 - 54 = 126^{\circ}\).
\(\angle5\) and \(\angle6\) are vertical angles. By the vertical - angles theorem (\(\angle5\cong\angle6\)), so \(m\angle6 = 126^{\circ}\).

Step3: Parallel - line and transversal (corresponding angles, alternate - interior angles, etc.)

Assume the relevant lines are parallel. \(\angle7\) and \(\angle1\) are corresponding angles. So \(m\angle7=m\angle1 = 36^{\circ}\).
\(\angle8\) and \(\angle4\) are corresponding angles. So \(m\angle8=m\angle4 = 54^{\circ}\).
\(\angle9\) and \(\angle5\) are corresponding angles. So \(m\angle9=m\angle5 = 126^{\circ}\).
\(\angle10\) and \(\angle7\) are vertical angles. So \(m\angle10=m\angle7 = 36^{\circ}\).
\(\angle11\) and \(\angle8\) are vertical angles. So \(m\angle11=m\angle8 = 54^{\circ}\).
\(\angle12\) and \(\angle4\) are vertical angles. So \(m\angle12=m\angle4 = 54^{\circ}\).
\(\angle13\) and \(\angle12\) are supplementary (\(\angle12+\angle13 = 180^{\circ}\)), so \(m\angle13=180 - 54=126^{\circ}\).
\(\angle14\) and \(\angle11\) are vertical angles. So \(m\angle14=m\angle11 = 54^{\circ}\).

Answer:

a. \(m\angle1 = 36^{\circ}\)
b. \(m\angle2 = 90^{\circ}\)
c. \(m\angle4 = 54^{\circ}\)
d. \(m\angle5 = 126^{\circ}\)
e. \(m\angle6 = 126^{\circ}\)
f. \(m\angle7 = 36^{\circ}\)
g. \(m\angle8 = 54^{\circ}\)
h. \(m\angle9 = 126^{\circ}\)
i. \(m\angle10 = 36^{\circ}\)
j. \(m\angle11 = 54^{\circ}\)
k. \(m\angle12 = 54^{\circ}\)
l. \(m\angle13 = 126^{\circ}\)
m. \(m\angle14 = 54^{\circ}\)