The following is an incomplete paragraph proving that the opposite angles of parallelogram abcd are congruent: according to the given information, and . using a straightedge, extend segment ab and place point p above point b. by the same reasoning, extend segment ad and place point t to the left of point a. angles bcd and pbc are congruent by the alternate interior angles theorem. angles pbc and bad are congruent by the by the transitive property of equality, angles bcd and bad are congruent. angles abc and bat are congruent by the angles bat and cda are congruent by the corresponding angles theorem. by the transitive property of equality, ∠abc is congruent to ∠cda. consequently, opposite angles of parallelogram abcd are congruent. what theorems accurately complete the proof?

Given a parallelogram ABCD, refer to the attached image.

To prove: Opposite angles are congruent.

Proof:

According to the given information,

Step 1: Using a straightedge, extend segment AB and place point P above point B.

Step 2: By the same reasoning, extend segment AD and place point T to the left of point A.

Step 3: Angles BCD and PBC are congruent by the Alternate Interior Angles Theorem.

Step 4: Angles PBC and BAD are congruent by the Corresponding Angles Theorem. ( If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Corresponding angles are the angles which occupy the same relative position at each intersection where a straight line crosses two others.)

Step 5: By the Transitive Property of Equality, angles BCD and BAD are congruent.

Step 6: Angles ABC and BAT are congruent by the Alternate Interior Theorem.(The Alternate Interior Angles theorem states, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alternate Interior angles are a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal.)

Step 7: Angles BAT and CDA are congruent by the Corresponding Angles Theorem.

Step 8: By the Transitive Property of Equality, ∠ABC is congruent to ∠CDA. Consequently, opposite angles of parallelogram ABCD are congruent.