Question

select the correct answer from each drop - down menu.
quadrilateral (abcd) is a parallelogram. complete the statements to prove that (overline{ab}congoverline{cd}) and (overline{bc}congoverline{ad}).

statement	reason
(overline{ab}paralleloverline{cd}) and (overline{bc}paralleloverline{ad})	definition of a parallelogram
draw (overline{ac}). (overline{ac}) is a transversal that intersects two pairs of parallel lines, (overline{ab}) and (overline{cd}) and (overline{ad}) and (overline{bc}).	drawing a line segment
(angle cabcongangle acd) and (angle bcacongangle cad)
(mangle cab = mangle acd) and (mangle bca=mangle cad)
(overline{ac}=overline{ac})
(\triangle abccong\triangle cda)	asa criterion for congruence
(overline{ab}congoverline{cd}) and (overline{bc}congoverline{ad})	corresponding sides of congruent triangles are congruent.

Explanation:

Step1: Identify angle relationship

$\angle CAB \cong \angle ACD$ and $\angle BCA \cong \angle CAD$ are formed by a transversal cutting parallel lines, these are alternate interior angles.

Step2: Match to correct theorem

The Alternate Interior Angles Theorem states that when a transversal intersects two parallel lines, alternate interior angles are congruent, which justifies the angle congruence statement.

Step3: Justify angle measure equality

If $\angle CAB \cong \angle ACD$ and $\angle BCA \cong \angle CAD$, then $m\angle CAB=m\angle ACD$ and $m\angle BCA=m\angle CAD$ because congruent angles have equal measures.

Step4: Justify shared side equality

$\overline{AC} = \overline{AC}$ is true by the reflexive property of congruence (a segment is congruent to itself).

Answer:

1. For $\angle CAB \cong \angle ACD$ and $\angle BCA \cong \angle CAD$: Alternate Interior Angles Theorem
2. For $m\angle CAB=m\angle ACD$ and $m\angle BCA=m\angle CAD$: Congruent angles have equal measures.
3. For $\overline{AC} = \overline{AC}$: Reflexive property of congruence