if angle 4 is 42 degrees and lines M and N are parallel, give a reason why angle 5 is also 42 degrees.
a. angle 4 and angle 5 are interior angles, so their measure must be the same.
b. angle 4 and angle 5 are corresponding angles, so their measures are congruent.
c. Angle 4 and angle 5 are vertical angles, so their measures are congruent.
d. angle 4 and angle 5 are alternate interior angles, so their measures must be the same
c. Angle 4 and angle 5 are vertical angles, so their measures are congruent.
The correct answer is option D. Angle 4 and angle 5 are alternate interior angles, so their measures must be the same.
To understand why angle 5 is also 42 degrees, let's first review the definitions of the different types of angles formed when two lines are intersected by a transversal:
1. Interior angles: These are the angles formed on the inside of the two intersecting lines.
2. Corresponding angles: These are the angles that are in the same relative position on each of the two intersecting lines.
3. Vertical angles: These are the pairs of angles opposite each other when two lines intersect. They have the same measure.
4. Alternate interior angles: These are the angles that are on the inside of the two lines but on opposite sides of the transversal.
In this scenario, we are given that angle 4 is 42 degrees and lines M and N are parallel. The angles formed by the intersection of lines M and N with the transversal are indicated as angle 4 and angle 5.
Based on the properties of angles formed when two parallel lines are intersected by a transversal, alternate interior angles are congruent. It means that if one angle is 42 degrees, then the other angle formed by the same pair of lines and the same transversal will also be 42 degrees.
Therefore, option D, which states that angle 4 and angle 5 are alternate interior angles, and their measures must be the same, is the correct reason why angle 5 is also 42 degrees.