Question

in the figure below, lines ( l ) and ( k ) are parallel.
suppose that ( mangle4 = 54^{circ} ) and ( mangle5 = 79^{circ} ).

complete the statements below.

by the angle addition property, ( mangle2 + mangle4 + mangle5 = square^{circ} ).
we are given that ( mangle4 = 54^{circ} ) and ( mangle5 = 79^{circ} ), so ( mangle2 = square^{circ} ).

we see that ( angle4 ) and ( angle1 ) are
and since lines ( l ) and ( k ) are parallel, ( angle4 ) and ( angle1 ) are
so, ( mangle1 = square^{circ} ).

we see that ( angle3 ) and ( angle5 ) are
and since lines ( l ) and ( k ) are parallel, ( angle3 ) and ( angle5 ) are
so, ( mangle3 = square^{circ} ).

therefore, ( mangle1 + mangle2 + mangle3 = square^{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 ( square^{circ} ).

Explanation:

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

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

Step2: Find \(m\angle2\)

Given \(m\angle4 = 54^{\circ}\) and \(m\angle5=79^{\circ}\).
Using \(m\angle2=180-(m\angle4 + m\angle5)\)
\(m\angle2=180-(54 + 79)=180 - 133=47^{\circ}\)

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, \(m\angle1 = 54^{\circ}\)

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, \(m\angle3 = 79^{\circ}\)

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

\(m\angle1 + m\angle2 + m\angle3=54 + 47+79 = 180^{\circ}\)

Answer:

By the angle addition property, \(m\angle2 + m\angle4 + m\angle5 = 180^{\circ}\).
We are given that \(m\angle4 = 54^{\circ}\) and \(m\angle5 = 79^{\circ}\), so \(m\angle2 = 47^{\circ}\).
We see that \(\angle4\) and \(\angle1\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle4\) and \(\angle1\) are congruent. So, \(m\angle1 = 54^{\circ}\).
We see that \(\angle3\) and \(\angle5\) are alternate interior angles. And since lines \(l\) and \(k\) are parallel, \(\angle3\) and \(\angle5\) are congruent. So, \(m\angle3 = 79^{\circ}\).
Therefore, \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\).
The relationship between \(\angle1,\angle2,\) and \(\angle3\) is an example of the rule: The sum of the interior angle measures of a triangle is \(180^{\circ}\).