Question

given abcd is a parallelogram
prove \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{bc} \cong \overline{da}\\)

there are two congruent triangles in the
diagram.

angles segments triangles statements reasons
asa cpctc
given reflexive property

statements reasons
1 abcd is a parallelogram 3. def. of parallelogram
2 draw \\(\overline{ac}\\) 4. reflexive property
3 \\(\overline{ab} \parallel \overline{cd}\\) 5. def. of alt. interior angles
4 \\(\overline{ac} \cong \overline{ac}\\) 6. def. of alt. interior angles
5 \\(\angle bca\\) and \\(\angle dac\\)
are alt. interior angles 7. def. of parallelogram
6 \\(\angle dca\\) and \\(\angle bac\\)
are alt. interior angles 8. alternate interior angles theorem
7 \\(\overline{bc} \parallel \overline{da}\\) 9. alternate interior angles theorem

Explanation:

Step1: Match given statement

Statement 1: $ABCD$ is a parallelogram
Reason: given

Step2: Match auxiliary line

Statement 2: draw $\overline{AC}$
Reason: (no pre-filled match, this is a construction step)

Step3: Match parallel sides def

Statement 3: $\overline{AB} \parallel \overline{CD}$
Reason: 3. def. of parallelogram

Step4: Match reflexive property

Statement 4: $\overline{AC} \cong \overline{AC}$
Reason: 4. reflexive property

Step5: Match alt. int. angles def

Statement 5: $\angle BCA$ and $\angle DAC$ are alt. interior angles
Reason: 5. def. of alt. interior angles

Step6: Match alt. int. angles def

Statement 6: $\angle DCA$ and $\angle BAC$ are alt. interior angles
Reason: 6. def. of alt. interior angles

Step7: Match parallel sides def

Statement 7: $\overline{BC} \parallel \overline{DA}$
Reason: 7. def. of parallelogram

Step8: Prove triangle congruence

Statement 8: $\triangle ABC \cong \triangle CDA$
Reason: ASA (using statements 5,4,6)

Step9: Apply CPCTC for segments

Statement 9: $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$
Reason: CPCTC

Answer:

Completed two-column proof:

Statements	Reasons
2. draw $\overline{AC}$	2. construction
3. $\overline{AB} \parallel \overline{CD}$	3. def. of parallelogram
4. $\overline{AC} \cong \overline{AC}$	4. reflexive property
5. $\angle BCA$ and $\angle DAC$ are alt. interior angles	5. def. of alt. interior angles
6. $\angle DCA$ and $\angle BAC$ are alt. interior angles	6. def. of alt. interior angles
7. $\overline{BC} \parallel \overline{DA}$	7. def. of parallelogram
8. $\angle BCA \cong \angle DAC$	8. alternate interior angles theorem
9. $\angle DCA \cong \angle BAC$	9. alternate interior angles theorem
10. $\triangle ABC \cong \triangle CDA$	10. ASA
11. $\overline{AB} \cong \overline{CD}$, $\overline{BC} \cong \overline{DA}$	11. CPCTC

Final proven result: $\boldsymbol{\overline{AB} \cong \overline{CD}}$ and $\boldsymbol{\overline{BC} \cong \overline{DA}}$