How Many Different Angle Measures Can Be Found Among The Eight Angles Formed When Two Parallel Lines Are Cut By A Transversal?

Answer: When two parallel lines are cut by a transversal, eight angles are formed. However, these eight angles consist of only two different angle measures if the lines are truly parallel.

Here is how we can determine this:

1. The angles directly across from each other when the lines are cut by the transversal are called vertical angles. Vertical angles are always equal. Therefore, every pair of vertical angles have the same measure.

2. The angles that are inside the parallel lines and on the same side of the transversal are called consecutive interior angles. These angles sum up to 180 degrees because the lines are parallel (this is called the consecutive interior angles theorem).

3. The angles that are outside the parallel lines and on the same side of the transversal are called consecutive exterior angles. These also sum up to 180 degrees for the same reason as consecutive interior angles.

4. The angles that are inside the parallel lines but on opposite sides of the transversal are called alternate interior angles. These are equal to each other due to the alternate interior angles theorem, which states that alternate interior angles are equal when the lines are parallel.

5. The angles that are outside the parallel lines and on opposite sides of the transversal are called alternate exterior angles. Like the alternate interior angles, these are also equal to each other for the same reason.

So, out of the eight angles:

- The four vertical angles form two pairs of equal angles. - The consecutive interior angles form two angles that add up to 180 degrees. - The consecutive exterior angles form two angles that also add up to 180 degrees. - Both pairs of alternate interior angles are equal to each other. - Both pairs of alternate exterior angles are equal to each other.

Since the consecutive interior and consecutive exterior angles add up to 180 degrees, and given that there are two sets of vertical angles and two sets of alternate angles (interior and exterior), it means that there are effectively two different angle measures among the eight angles: one angle measure that is less than 180 degrees and its supplementary angle that is the difference between 180 degrees and the first angle measure.