Question

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

complete the statements below.

by the angle addition property, ( mangle 4 + mangle 2 + mangle 5 = ) ( ^{circ} ).
we are given that ( mangle 4 = 43^{circ} ) and ( mangle 5 = 72^{circ} ), so ( mangle 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, ( mangle 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, ( mangle 3 = ) ( ^{circ} ).

therefore, ( mangle 1 + mangle 2 + mangle 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: Find \(m\angle4 + m\angle2 + m\angle5\)

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

Step2: Calculate \(m\angle2\)

Given \(m\angle4 = 43^{\circ}\) and \(m\angle5 = 72^{\circ}\). Substitute into \(m\angle4 + m\angle2 + m\angle5 = 180^{\circ}\)
\(m\angle2=180-(m\angle4 + m\angle5)\)
\(m\angle2=180-(43 + 72)\)
\(m\angle2=180 - 115\)
\(m\angle2 = 65^{\circ}\)

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

We see that \(\angle4\) and \(\angle1\) are alternate interior angles (form a 'Z' - like shape). And since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle1\) are congruent (by the Alternate Interior Angles Theorem). So, \(m\angle1=m\angle4 = 43^{\circ}\)

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

We see that \(\angle3\) and \(\angle5\) are alternate interior angles (form a 'Z' - like shape). And since lines \(l\) and \(k\) are parallel, \(\angle3\) and \(\angle5\) are congruent (by the Alternate Interior Angles Theorem). So, \(m\angle3=m\angle5 = 72^{\circ}\)

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

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

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 \(180^{\circ}\).