Consider the figure below.  If a transversal line intersects two parallel lines, then the two interior angles on the same side of the transversal (the two consecutive, interior angles of the transversal) are supplementary.  This means that they add up to be 180.

The Corresponding Angles Theorem states that if two parallel lines are cut by a transversal line then the pair of corresponding angles are congruent.  Corresponding angles are angles formed when a transversal line cuts two lines and they lie in the same position at each intersection.  The figure below illustrates corresponding lines.

The correct answer is "If two parallel lines are cut by a transversal line, the alternate interior angles are congruent."

Alternate interior angles are pairs of angles on the inner sides of parallel lines but on the opposite sides of a transversal line which is intersecting parallel lines.  These angles are always congruent.  Alternate interior angles are illustrated in the figure below.

This is true according to the Alternate Exterior Angle Theorem which can be proven in a similar way to the Alternate Interior Angle Theorem.  Alternate exterior angles are illustrated in the figure below.

The definition of complementary angles is: any two angles that sum to 90.  We most often see these angles as the two angles in a right triangle that are not the right angle.  These two angles do not have to only be in right triangles, however.  Complementary triangles are any pair of angles that add up to be 90.

We must use the fact that lines  and are parallel lines to solve for the missing angles.  We will break it down to solve for each angle one at a time.

Angle :

We know that angle ’s supplementary angle.  Supplementary angles are two angles that add up to 180 degrees.  These two are supplementary angles because they form a straight line and straight lines are always 180 degrees.  So to solve for angle we simply subtract it’s supplementary angle from 180.

Angle :

We now know that angle  is 130 degrees.  We can either use that fact that angles  and  are opposite vertical angles to find the value of angle  or we can use that fact that angle ’s supplementary angle is the given angle of 50 degrees.  If we use the latter, we would use the same procedure as last time to solve for angle .  If we use the fact that angles  and  are opposite vertical angles, we know that they are congruent.  Since angle  then angle .

Angle :

To find angle  we can use the fact that angles  and  are corresponding angles and therefore are congruent or we can use the fact that angles  and  are alternate interior angles and therefore are congruent.  Either method that we use will show that .

Angle :

To find angle  we can use that fact that the given angle of 50 degrees and angle  are alternate exterior angles and therefore are congruent, or we can use the fact that angle  is angle 's supplementary angle.  We know that the given angle and angle  are alternate exterior angles so .

We know that lines AB and CD are parallel due to the information we get from the angles formed by the transversal line.  There are:

• two pairs of congruent vertically opposite angles
• two pairs of congruent alternate interior angles
• two pairs of congruent alternate exterior angles
• two pairs of congruent corresponding angles

Just using any one of these facts is enough proof that lines AB and CD are parallel.

Even though it may not be obvious at first, the given angle is actually a supplementary angle to angle .  This is because the given angle is corresponding (and therefore congruent) angles to the angle adjacent to angle .  Since they are supplementary we can set up the following equation.

To answer this question, we must understand the definition of vertically opposite angles.  Vertically opposite angles are angles that are formed opposite of each other when two lines intersect.  Vertically opposite angles are always congruent to each other.

Follow the detailed proof below.