Question

1. if ( m angle 3 = 54 ^ { circ } ), find each measure.

a. ( m angle 1 = ) b. ( m angle 2 = ) c. ( m angle 4 = ) d. ( m angle 5 = ) e. ( m angle 6 = )
f. ( m angle 7 = ) g. ( m angle 8 = ) h. ( m angle 9 = ) 1. ( m angle 10 = ) j. ( m angle 11 = )
k. ( m angle 12 = ) 1. ( m angle 13 = ) m. ( m angle 14 = )

Explanation:

Step1: Vertical angles

Vertical angles are equal. So, \(m\angle3 = m\angle5=54^{\circ}\) (vertical angles).

Step2: Complementary angles

Since \(\angle1\) and \(\angle3\) are complementary (\(\angle1+\angle3 = 90^{\circ}\)), then \(m\angle1=90^{\circ}-m\angle3 = 90^{\circ}- 54^{\circ}=36^{\circ}\).

Step3: Right angle

\(\angle2\) is a right - angle, so \(m\angle2 = 90^{\circ}\).

Step4: Vertical angles and linear pairs

\(m\angle4=m\angle1 = 36^{\circ}\) (vertical angles).
\(m\angle6 = 180^{\circ}-m\angle5=180^{\circ}-54^{\circ}=126^{\circ}\) (linear pair).
\(m\angle7=m\angle6 = 126^{\circ}\) (vertical angles).
\(m\angle8=m\angle5 = 54^{\circ}\) (vertical angles).
\(m\angle9=m\angle6 = 126^{\circ}\) (vertical angles).
\(m\angle10=m\angle7 = 126^{\circ}\) (vertical angles).
\(m\angle11=m\angle8 = 54^{\circ}\) (vertical angles).
\(m\angle12=m\angle4 = 36^{\circ}\) (vertical angles).
\(m\angle13=m\angle12 = 36^{\circ}\) (vertical angles).
\(m\angle14=m\angle11 = 54^{\circ}\) (vertical angles).

Answer:

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