Question

c. are any pairs of angles in your diagram congruent? explain your reasoning.
are any pairs of angles supplementary? explain your reasoning.
turn and talk: what do you think will happen with the angle relationships if the two lines crossed by the transversal are parallel?

Explanation:

Step1: Define congruent angles

Congruent angles have equal measures. In a diagram with a transversal crossing lines, vertical - angles are always congruent. For example, if we have two intersecting lines, the angles opposite each other are vertical angles and are congruent. Also, if the two lines crossed by the transversal are parallel, corresponding angles (angles in the same relative position) are congruent, alternate - interior angles (between the two lines and on opposite sides of the transversal) are congruent, and alternate - exterior angles (outside the two lines and on opposite sides of the transversal) are congruent.

Step2: Define supplementary angles

Supplementary angles are two angles whose sum is 180 degrees. In a diagram with a transversal, if two adjacent angles form a straight - line, they are supplementary. Also, if the two lines crossed by the transversal are parallel, same - side interior angles (between the two lines and on the same side of the transversal) are supplementary and same - side exterior angles (outside the two lines and on the same side of the transversal) are supplementary.

Step3: Analyze parallel lines case

If the two lines crossed by the transversal are parallel:

• Congruent angles: Corresponding angles are congruent, alternate - interior angles are congruent, and alternate - exterior angles are congruent.
• Supplementary angles: Same - side interior angles are supplementary and same - side exterior angles are supplementary.

Answer:

For congruent angles: In general, vertical angles are congruent. If the lines crossed by the transversal are parallel, corresponding, alternate - interior, and alternate - exterior angles are congruent.
For supplementary angles: Adjacent angles forming a straight - line are supplementary. If the lines crossed by the transversal are parallel, same - side interior and same - side exterior angles are supplementary.
If the two lines crossed by the transversal are parallel, corresponding, alternate - interior, and alternate - exterior angles are congruent, and same - side interior and same - side exterior angles are supplementary.