Question

in the figure below, lines ( l ) and ( k ) are parallel. suppose that ( mangle 4 = 36^{circ} ) and ( mangle 2 = 48^{circ} ). complete the statements below. we see that ( angle 1 ) and ( angle 4 ) are and since lines ( l ) and ( k ) are parallel, ( angle 1 ) and ( angle 4 ) are so, ( mangle 1 = ) ( ^{circ} ). we see that ( angle 2 ) and ( angle 5 ) are and since lines ( l ) and ( k ) are parallel, ( angle 2 ) and ( angle 5 ) are so, ( mangle 5 = ) ( ^{circ} ). by the angle addition property, ( mangle 5 + mangle 4 + mangle 3 = ) ( ^{circ} ). note that ( mangle 5 = 48^{circ} ) and ( mangle 4 = 36^{circ} ), so ( mangle 3 = ) ( ^{circ} ).

Explanation:

Step1: Determine the relationship between ∠1 and ∠4

∠1 and ∠4 are corresponding angles.
Since lines \(l\) and \(k\) are parallel, by the Corresponding Angles Postulate, ∠1 and ∠4 are congruent.
So, \(m\angle1 = 36^{\circ}\)

Step2: Determine the relationship between ∠2 and ∠5

∠2 and ∠5 are alternate interior angles.
Since lines \(l\) and \(k\) are parallel, by the Alternate Interior Angles Theorem, ∠2 and ∠5 are congruent.
So, \(m\angle5 = 48^{\circ}\)

Step3: Use the angle - addition property

By the angle - addition property, \(m\angle5 + m\angle4 + m\angle3=180^{\circ}\) (because they form a straight angle).
Given \(m\angle5 = 48^{\circ}\) and \(m\angle4 = 36^{\circ}\), we substitute these values into the equation:
\(48^{\circ}+36^{\circ}+m\angle3 = 180^{\circ}\)
\(m\angle3=180^{\circ}-(48^{\circ} + 36^{\circ})\)
\(m\angle3=180^{\circ}-84^{\circ}\)
\(m\angle3 = 96^{\circ}\)

Answer:

\(m\angle1 = 36^{\circ}\), \(m\angle5 = 48^{\circ}\), \(m\angle5 + m\angle4 + m\angle3=180^{\circ}\), \(m\angle3 = 96^{\circ}\)