Question

14
what are the missing statement and reason in step 2 of the proof?
proof:
statements

1. quadrilateral abcd is a parallelogram.
2. ?
3. draw \\( \overline { a c } \\), a transversal that cuts two pairs of parallel segments. \\( \frac { \overline { a b } \parallel \overline { c d } } { \overline { b c } \parallel \overline { a d } } \\)
4. \\( \angle c a b \cong \angle a c d \\) \\( \angle b c a \cong \angle c a d \\)
5. \\( m \angle c a b = m \angle a c d \\) \\( m \angle b c a = m \angle c a d \\)
6. \\( a c = a c \\)
7. ?
8. \\( \frac { \overline { a b } \cong \overline { c d } } { \overline { b c } \cong \overline { a d } } \\)

reasons
given
drawing a line segment
alternate interior angles theorem
definition of congruent
reflexive property of equality
asa criterion for congruence
corresponding sides of congruent triangles are congruent
a. statement: \\( \overline { a b } \cong \overline { c d } , \overline { b c } \cong \overline { a d } \\)
reason: given
b. statement: \\( \triangle a b c \cong \triangle c d a \\)
reason: sss criterion for congruence
c. statement: \\( \overline { a b } \parallel \overline { c d } , \overline { b c } \parallel \overline { a d } \\)
reason: definition of a parallelogram
d. statement: \\( \angle b a c \cong \angle a c d \\)
reason: alternate interior angles theorem

Explanation:

• A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel.
• In step 1, it is given that \(ABCD\) is a parallelogram. So, by the definition of a parallelogram, the opposite sides are parallel. That is \(AB\parallel CD\) and \(BC\parallel AD\).

Answer:

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