Question

in the figure below, lines ( l ) and ( k ) are parallel.
suppose that ( m angle 4 = 43^{circ} ) and ( m angle 5 = 72^{circ} ).

complete the statements below.

by the angle addition property, ( m angle 4 + m angle 2 + m angle 5 = ) ( ^{circ} ).
we are given that ( m angle 4 = 43^{circ} ) and ( m angle 5 = 72^{circ} ), so ( m angle 2 = ) ( ^{circ} ).

we see that ( angle 4 ) and ( angle 1 ) are
and since lines ( l ) and ( k ) are parallel, ( angle 4 ) and ( angle 1 ) are
so, ( m angle 1 = ) ( ^{circ} ).

we see that ( angle 3 ) and ( angle 5 ) are
and since lines ( l ) and ( k ) are parallel, ( angle 3 ) and ( angle 5 ) are
so, ( m angle 3 = ) ( ^{circ} ).

therefore, ( m angle 1 + m angle 2 + m angle 3 = ) ( ^{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 ( ^{circ} ).

Explanation:

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

By the angle addition property, \(m\angle4 + m\angle2 + m\angle5=180^{\circ}\) (since they form a straight - line angle).

Step2: Find \(m\angle2\)

We know that \(m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\), and \(m\angle4 = 43^{\circ}\), \(m\angle5=72^{\circ}\).
Substitute the values: \(43^{\circ}+m\angle2 + 72^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(m\angle2+115^{\circ}=180^{\circ}\).
Subtract \(115^{\circ}\) from both sides: \(m\angle2=180^{\circ}-115^{\circ}=65^{\circ}\).

Step3: Analyze \(\angle4\) and \(\angle1\)

We see that \(\angle4\) and \(\angle1\) are alternate interior angles.
Since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle1\) are congruent.
So, \(m\angle1 = m\angle4=43^{\circ}\).

Step4: Analyze \(\angle3\) and \(\angle5\)

We see that \(\angle3\) and \(\angle5\) are alternate interior angles.
Since lines \(l\) and \(k\) are parallel, \(\angle3\) and \(\angle5\) are congruent.
So, \(m\angle3 = m\angle5 = 72^{\circ}\).

Step5: Calculate \(m\angle1 + m\angle2 + m\angle3\)

Substitute \(m\angle1 = 43^{\circ}\), \(m\angle2 = 65^{\circ}\), \(m\angle3 = 72^{\circ}\) into \(m\angle1 + m\angle2 + m\angle3\).
\(m\angle1 + m\angle2 + m\angle3=43^{\circ}+65^{\circ}+72^{\circ}=180^{\circ}\).

Answer:

By the angle addition property, \(m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\).
We are given that \(m\angle4 = 43^{\circ}\) and \(m\angle5 = 72^{\circ}\), so \(m\angle2 = 65^{\circ}\).
We see that \(\angle4\) and \(\angle1\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle1\) are congruent. So, \(m\angle1 = 43^{\circ}\).
We see that \(\angle3\) and \(\angle5\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle3\) and \(\angle5\) are congruent. So, \(m\angle3 = 72^{\circ}\).
Therefore, \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\).
The relationship between \(\angle1\), \(\angle2\), and \(\angle3\) is an example of the following rule. The sum of the interior angle measures of a triangle is \(180^{\circ}\).