Question

given the diagram and ( m||n ), determine the measure of each angle.
( mangle1=) (circ)
( mangle2=) (circ)
( mangle3=) (circ)
( mangle4=) (circ)
( mangle5=) (circ)
( mangle6=) (circ)
( mangle7=) (circ)
( mangle8=) (circ)
( mangle9=) (circ)
( mangle10=) (circ)

Explanation:

Step1: Vertical angles

Vertical angles are equal. So \(m\angle3 = 47^{\circ}\) (since \(\angle3\) and the \(47^{\circ}\) angle are vertical angles).

Step2: Linear - pair angles

\(\angle1\) and the \(47^{\circ}\) angle form a linear pair. So \(m\angle1=180 - 47=133^{\circ}\).
Since \(\angle2\) and \(\angle1\) are vertical angles, \(m\angle2 = 47^{\circ}\).

Step3: Right - angle and complementary angles

The right - angle is \(90^{\circ}\). \(\angle4\) and \(\angle5\) are complementary. \(\angle1\) and \(\angle5\) are supplementary with the right - angle. \(m\angle5=180-(90 + 47)=43^{\circ}\), \(m\angle4 = 90 - 43=47^{\circ}\). \(m\angle6=47^{\circ}\) (vertical angle of \(\angle4\)).

Step4: Parallel lines and corresponding angles

Since \(m\parallel n\), \(\angle7\) and \(\angle2\) are vertical angles, \(m\angle7 = 47^{\circ}\). \(\angle8\) and \(\angle5\) are corresponding angles, \(m\angle8 = 43^{\circ}\). \(\angle9\) and \(\angle5\) are vertical angles (by the property of parallel lines and transversals), \(m\angle9 = 43^{\circ}\). \(\angle10\) and \(\angle7\) are vertical angles, \(m\angle10 = 47^{\circ}\).

Answer:

\(m\angle1 = 133^{\circ}\)
\(m\angle2 = 47^{\circ}\)
\(m\angle3 = 47^{\circ}\)
\(m\angle4 = 47^{\circ}\)
\(m\angle5 = 43^{\circ}\)
\(m\angle6 = 47^{\circ}\)
\(m\angle7 = 47^{\circ}\)
\(m\angle8 = 43^{\circ}\)
\(m\angle9 = 43^{\circ}\)
\(m\angle10 = 47^{\circ}\)