Question

geometry 3.3 ticket
given: ( m parallel n ). identify the measure and relationship of all angles starting with ( mangle1 = 140^{circ} ) then state the relationship to ( angle1 ) if there is one.
( mangle2=)__ relationship:__
( mangle3=)__ relationship:__
( mangle4=)__ relationship:__
( mangle5=)__ relationship:__
( mangle6=)__ relationship:__
( mangle7=)__ relationship:__
( mangle8=)__ relationship:__

Explanation:

Step1: Find \(m\angle2\)

\(\angle1\) and \(\angle2\) are supplementary (\(\angle1+\angle2 = 180^{\circ}\)).
\(m\angle2=180^{\circ}-m\angle1 = 180^{\circ}- 140^{\circ}=40^{\circ}\)
Relationship: Supplementary angles.

Step2: Find \(m\angle3\)

\(\angle1\) and \(\angle3\) are vertical angles (\(\angle1\cong\angle3\)).
\(m\angle3 = m\angle1=140^{\circ}\)
Relationship: Vertical angles.

Step3: Find \(m\angle4\)

\(\angle2\) and \(\angle4\) are vertical angles (\(\angle2\cong\angle4\)).
\(m\angle4=m\angle2 = 40^{\circ}\)
Relationship: Vertical angles (also supplementary to \(\angle1\) as \(\angle1+\angle4=180^{\circ}\)).

Step4: Find \(m\angle5\)

\(\angle1\) and \(\angle5\) are alternate - exterior angles (\(m\parallel n\), so \(\angle1\cong\angle5\)).
\(m\angle5 = m\angle1=140^{\circ}\)
Relationship: Alternate - exterior angles.

Step5: Find \(m\angle6\)

\(\angle5\) and \(\angle6\) are supplementary (\(\angle5+\angle6 = 180^{\circ}\)).
\(m\angle6=180^{\circ}-m\angle5 = 180^{\circ}-140^{\circ}=40^{\circ}\)
Relationship: Supplementary angles (also corresponding to \(\angle2\)).

Step6: Find \(m\angle7\)

\(\angle5\) and \(\angle7\) are vertical angles (\(\angle5\cong\angle7\)).
\(m\angle7=m\angle5 = 140^{\circ}\)
Relationship: Vertical angles (also corresponding to \(\angle3\)).

Step7: Find \(m\angle8\)

\(\angle6\) and \(\angle8\) are vertical angles (\(\angle6\cong\angle8\)).
\(m\angle8=m\angle6 = 40^{\circ}\)
Relationship: Vertical angles (also corresponding to \(\angle4\)).

Answer:

\(m\angle2 = 40^{\circ}\), Relationship: Supplementary angles.\(m\angle3 = 140^{\circ}\), Relationship: Vertical angles.\(m\angle4 = 40^{\circ}\), Relationship: Vertical angles (supplementary to \(\angle1\)).\(m\angle5 = 140^{\circ}\), Relationship: Alternate - exterior angles.\(m\angle6 = 40^{\circ}\), Relationship: Supplementary angles (corresponding to \(\angle2\)).\(m\angle7 = 140^{\circ}\), Relationship: Vertical angles (corresponding to \(\angle3\)).\(m\angle8 = 40^{\circ}\), Relationship: Vertical angles (corresponding to \(\angle4\)).