Question

Match the following reasons with the statements given to create the proof.

Given: ABCD is a quadrilateral
DO-OB
AO-OC
Prove: ABCD is a parallelogram
1. Moverline[DO)-\overline(OB),\ \overline[AO]=\overline(OC))
2. DOC=AOB
3. Triangle COD congruent to Triangle AOB
4. 21-22. overline[AB]=\overline(CD))
5. Moverline(DC\\\\parallel\ \overline[AB])
6. ABCD is a parallelogram
If alternate interior angles, then
lines parallel.
SAS
Given
CPCTE
Vertical angles are equal.
If two sides and II, then a
parallelogram.

Answer 1

The matching of the reasons with the statements to create the proof:

DO=OB Given

AO=OC Given

DOC=AOB Vertical angles are equal.

Triangle COD congruent to Triangle AOB 3. SAS

21-22. AB=CD CPCTE

`Moverline(DC\parallel\ AB)` If two sides and II, then a parallelogram

ABCD is a parallelogram If alternate interior angles, then lines parallel.

Vertical angles are equal: Angles DOC and AOB are vertical angles, so they are congruent (equal).

SAS: Since DO = OB, AO = OC, and DOC = AOB, triangle COD is congruent to triangle AOB by the SAS postulate.

CPCTE: Corresponding parts of congruent triangles are equal. Therefore, AB = CD (corresponding sides).

If two sides and II, then a parallelogram: If a quadrilateral has two sides parallel and congruent, it is a parallelogram.

If alternate interior angles, then lines parallel: Since AB is parallel to CD, and DC is a transversal, alternate interior angles DOC and AOB are congruent.

Conclusion: Since both conditions for a parallelogram are met (two sides parallel and congruent and alternate interior angles congruent), ABCD is proven to be a parallelogram.