Question

we see that \\( \angle 5 \\) and \\( \angle 2 \\) are vertical angles
and since lines \\( l \\) and \\( k \\) are parallel, \\( \angle 5 \\) and \\( \angle 2 \\) are congruent
so, \\( m \angle 2 = 32 ^ { \circ } \\).
by the angle addition property, \\( m \angle 4 + m \angle 1 + m \angle 5 = 180 ^ { \circ } \\).
we are given that \\( m \angle 5 = 32 ^ { \circ } \\) and \\( m \angle 1 = 104 ^ { \circ } \\), so \\( m \angle 4 = 44 ^ { \circ } \\).
we see that \\( \angle 3 \\) and \\( \angle 4 \\) are alternate interior angles
and since lines \\( l \\) and \\( k \\) are parallel, \\( \angle 3 \\) and \\( \angle 4 \\) are congruent
so, \\( m \angle 3 = 44 ^ { \circ } \\).
therefore, \\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\).
the relationship between \\( \angle 1, \angle 2 \\), and \\( \angle 3 \\) is an example of the following rule.
the sum of the interior angle measures of a triangle is 180

Explanation:

Step1: Vertical Angles

Vertical angles are equal.

Step2: Angle Addition Property

\(m\angle4 + m\angle1 + m\angle5=180^{\circ}\), substitute \(m\angle5 = 32^{\circ}\) and \(m\angle1 = 104^{\circ}\)
\(m\angle4=180^{\circ}-32^{\circ}-104^{\circ}=44^{\circ}\)

Step3: Alternate Interior Angles

Alternate interior angles are equal when lines are parallel.

Step4: Triangle Angle Sum

Sum of interior angles of a triangle is \(180^{\circ}\)

Answer:

\(m\angle2 = 32^{\circ}\), \(m\angle4 = 44^{\circ}\), \(m\angle3 = 44^{\circ}\), sum of interior angles of a triangle \(180^{\circ}\)