Question

given: quadrilateral abcd is a parallelogram.
prove: (overline{ab}congoverline{cd})
(overline{bc}congoverline{ad})

proof:

statements	reasons
?
draw (overline{ac}), a transversal that cuts two pairs of parallel segments (overline{ab}paralleloverline{cd})(overline{bc}paralleloverline{ad})	drawing a line segment
(angle cabcongangle acd)(angle bcacongangle cad)	alternate interior angles theorem
(mangle cab = mangle acd)(mangle bca = mangle cad)	definition of congruent
(ac = ac)	reflexive property of equality
?	asa criterion for congruence
(overline{ab}congoverline{cd})(overline{bc}congoverline{ad})	corresponding sides of congruent triangles are congruent

what are the missing statement and reason in step 2 of the proof?

• a. statement: (\triangle abccong\triangle cda) reason: sss criterion for congruence
• b. statement: (angle baccongangle acd) reason: alternate interior angles theorem
• c. statement: (overline{ab}congoverline{cd},overline{bc}congoverline{ad}) reason: given
• d. statement: (overline{ab}paralleloverline{cd},overline{bc}paralleloverline{ad}) reason: definition of a parallelogram

Explanation:

A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel. In the proof, since we start with the given that \(ABCD\) is a parallelogram, the next logical step (step 2) is to state the property of a parallelogram regarding the parallelism of its sides. The definition of a parallelogram directly gives that \(\overline{AB}\parallel\overline{CD}\) and \(\overline{BC}\parallel\overline{AD}\).

Answer:

D. Statement: \(\overline{AB}\parallel\overline{CD},\overline{BC}\parallel\overline{AD}\) Reason: definition of a parallelogram