Question

the figure below, lines ( l ) and ( k ) are parallel.
ppose that ( m angle 4 = 38 ^ { circ } ) and ( m angle 5 = 34 ^ { circ } ).

complete the statements below.

by the angle addition property, ( m angle 2 + m angle 4 + m angle 5 = square ^ { circ } ).
we are given that ( m angle 4 = 38 ^ { circ } ) and ( m angle 5 = 34 ^ { circ } ), so ( m angle 2 = square ^ { circ } ).

we see that ( angle 4 ) and ( angle 3 ) are
and since lines ( l ) and ( k ) are parallel, ( angle 4 ) and ( angle 3 ) are
so, ( m angle 3 = square ^ { circ } ).

we see that ( angle 1 ) and ( angle 5 ) are
and since lines ( l ) and ( k ) are parallel, ( angle 1 ) and ( angle 5 ) are

Explanation:

Step1: Calculate \(m\angle2 + m\angle4 + m\angle5\)

By the angle addition property (a straight - line has an angle measure of \(180^{\circ}\)), \(m\angle2 + m\angle4 + m\angle5=180^{\circ}\)

Step2: Find \(m\angle2\)

We know that \(m\angle2 + m\angle4 + m\angle5 = 180^{\circ}\), and \(m\angle4 = 38^{\circ}\), \(m\angle5=34^{\circ}\).
Substitute the values into the equation: \(m\angle2=180-(m\angle4 + m\angle5)\)
\(m\angle2=180-(38 + 34)\)
\(m\angle2=180 - 72\)
\(m\angle2 = 108^{\circ}\)

Step3: Analyze \(\angle4\) and \(\angle3\)

We see that \(\angle4\) and \(\angle3\) are alternate interior angles.
Since lines \(l\) and \(k\) are parallel, by the alternate - interior angles theorem (if two parallel lines are cut by a transversal, then alternate interior angles are congruent), \(\angle4\) and \(\angle3\) are congruent.
So, \(m\angle3=m\angle4 = 38^{\circ}\)

Step4: Analyze \(\angle1\) and \(\angle5\)

We see that \(\angle1\) and \(\angle5\) are alternate interior angles.
Since lines \(l\) and \(k\) are parallel, by the alternate - interior angles theorem, \(\angle1\) and \(\angle5\) are congruent

Answer:

By the angle addition property, \(m\angle2 + m\angle4 + m\angle5 = 180^{\circ}\).
We are given that \(m\angle4 = 38^{\circ}\) and \(m\angle5 = 34^{\circ}\), so \(m\angle2=108^{\circ}\).
We see that \(\angle4\) and \(\angle3\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle3\) are congruent. So, \(m\angle3 = 38^{\circ}\).
We see that \(\angle1\) and \(\angle5\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle1\) and \(\angle5\) are congruent.