Question

The following is an incomplete paragraph proving that the opposite angles of parallelogram ABCD are congruent: Parallelogram ABCD is shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD. According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD. . 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? (5 points) 1. Corresponding Angles Theorem 2. Alternate Interior Angles Theorem 1. Alternate Interior Angles Theorem 2. Corresponding Angles Theorem 1. Corresponding Angles Theorem 2. Corresponding Angles Theorem 1. Alternate Interior Angles Theorem 2. Alternate Interior Angles Theorem

Answer 1

General Idea:

The angles which occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are equal.

The angles that are formed on opposite sides of the transversal and inside the two lines are alternate interior angles. The theorem says that when the lines are parallel, that the alternate interior angles are equal.

Applying the concept:

Angles PBC and BAD are congruent by the Corresponding Angle Theorem.

Angles ABC and BAT are congruent by the Alternate Interior angle Theorem.

Answer 2

According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD.. 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 Corresponding Angles Theorem. By the __________ Property of Equality, angles BCD and BAD are congruent. Angles ABC and BAT are congruent by the Alternate Interior Theorem. Angles BAT and CDA are congruent by the Corresponding Angles Theorem. By the __________ Property of Equality, ∠ABC is congruent to ∠CDA. Consequently, opposite angles of parallelogram ABCD are congruent.

the answer is that angles PBC and BAD are congruent by the corresponding angles theorem. Angles BAT and ABC are congruent by the Alternative Interior theorem. The answers are in bold.