1. Using Special Quadrilaterals
PROOFS
Using Special Quadrilaterals

2. #1 If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Given: PQRS is a parallelogram
Prove: 𝑃𝑄≅𝑅𝑆, 𝑄𝑅≅𝑃𝑆
Statements
Reasons
1. PQRS is a parallelogram.
1. Given
2. Through any 2 points there exists exactly 1 line.
2. Draw QS.
3. 𝑃𝑄∥𝑅𝑆, 𝑄𝑅∥𝑃𝑆
3. Definition of parallelogram.
4. ∠𝑃𝑄𝑆≅∠𝑅𝑆𝑄, ∠𝑃𝑆𝑄≅∠𝑅𝑄𝑆
4. Alternate Interior Angles Theorem.
5. 𝑄𝑆≅𝑄𝑆
5. Reflexive Property of Congruence.
6. ∆𝑃𝑄𝑆≅∆𝑅𝑆𝑄
6. ASA Congruence Postulate.
7. 𝑃𝑄≅𝑅𝑆, 𝑄𝑅≅𝑃𝑆
7. Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

3. #2 If both pairs of opposite sides in a quadrilateral are congruent, then it is a parallelogram.
Given: 𝑃𝑄≅𝑅𝑆, 𝑄𝑅≅𝑃𝑆
Prove: PQRS is a parallelogram
Statements
Reasons
1. 𝑃𝑄≅𝑅𝑆, 𝑄𝑅≅𝑃𝑆
1. Given
2. Through any 2 points there exists exactly 1 line.
2. Draw QS.
3. Reflexive Property of Congruence.
3. 𝑄𝑆≅𝑄𝑆
4. SSS Congruence Postulate.
4. ∆𝑃𝑄𝑆≅∆𝑅𝑆𝑄
5. ∠𝑃𝑄𝑆≅∠𝑅𝑆𝑄, ∠𝑃𝑆𝑄≅∠𝑅𝑄𝑆
5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
6. 𝑃𝑄∥𝑅𝑆, 𝑄𝑅∥𝑃𝑆
6. Alternate Interior Angles Theorem.
7. PQRS is a parallelogram.
7. Definition of parallelogram.

4. #3 If both pairs of opposite angles in a quadrilateral are congruent, then it’s a parallelogram.
Given: PQRS is a quadrilateral in which ∠𝑃≅∠𝑅, ∠𝑄≅∠𝑆
Prove: 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
Statements
Reasons
PQRS is a quadrilateral in which
∠𝑃≅∠𝑅, ∠𝑄≅∠𝑆
1. Given
2. The interior angle sum of a quadrilateral is 360.
2. 2x + 2y = 360
3. Multiplication property of equality.
3. 𝑥+𝑦=180
4. ∠𝑃,∠𝑄 𝑎𝑛𝑑 ∠𝑄,∠𝑅 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
4. Definition of supplementary angles
5. Consecutive Interior Angles Converse.
5. P𝑄∥𝑅𝑆, 𝑄𝑅 ∥𝑃𝑆
6. 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
6. Definition of Parallelogram

5. #4 If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Given: PQRS is a parallelogram
Prove:
𝑀 bisects 𝑄𝑆 and 𝑃𝑅
Statements
Reasons
1. Given
2. If a quadrilateral is a parallelogram, then its opposite sides are congruent.
3. Alternate Interior Angles Congruence Theorem
4. ASA
5. Corr. parts of ≅ triangles are ≅.
6. Definitions of segment bisector.

6. Statements Justifications Given
#5 Converse of Diagonals Bisect Each Other
Thm If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Statements Justifications
Given
Def. of Segment Bisector
Def. of Vertical Angles
SAS
CPCTC

7. #2 If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Given: PQRS is a parallelogram
Prove: ∠𝑃≅∠𝑅, ∠𝑄≅∠𝑆
Statements
Reasons
1. PQRS is a parallelogram.
1. Given
2. Through any 2 points there exists exactly 1 line.
2. Draw QS.
3. 𝑃𝑄∥𝑅𝑆, 𝑄𝑅∥𝑃𝑆
3. Reflexive Property of Congruence.
4. ∠𝑃𝑄𝑆≅∠𝑅𝑆𝑄, ∠𝑃𝑆𝑄≅∠𝑅𝑄𝑆
4. SSS
5. 𝑄𝑆≅𝑄𝑆
5. Corresponding Parts of Congruent Triangles are Congruent (CPCTC). .
6. ∆𝑃𝑄𝑆≅∆𝑅𝑆𝑄
6. Alternate Interior Angles Converse.
7.∠𝑃≅∠𝑅, ∠𝑄≅∠𝑆
7. Definition of Parallelogram