Math

2. Theorem 1: If a quadrilateral is a parallelogram, then the two
pairs of opposite sides are congruent.
Theorem 2: If a quadrilateral is a parallelogram, then the two
pairs of opposite angles are congruent.
Theorem 3: If a quadrilateral is a parallelogram, then the
consecutive angles are supplementary.
Theorem 4: If the diagonals of the quadrilateral bisect each
other, then the quadrilateral is a parallelogram.
Theorem 5: If a quadrilateral is a parallelogram, then
diagonals form two congruent triangles.

3. Properties of a Parallelogram

4. Statements Reasons
𝐴𝐵 || 𝐶𝐷, 𝐴𝐷 || 𝐵𝐶 Given
𝐴𝐶 is a diagonal
Definition of
diagonal
∠𝐵𝐴𝐶 ≅ ∠𝐷𝐶𝐴,
∠𝐷𝐴𝐶 ≅ ∠𝐵𝐶𝐴
Alternate Interior
Angle are congruent
𝐴𝐶 ≅ 𝐴𝐶 Reflexive Property
⊿𝐷𝐴𝐶 ≅ ⊿𝐵𝐶𝐴 ASA Postulate
𝐴𝐵 ≅ 𝐶𝐷 , 𝐴𝐷 ≅ 𝐵𝐶
Corresponding
parts of congruent
triangles are
congruent. (CPCTC)
PROOF

5. Theorem 1: If a quadrilateral is a parallelogram, then
the two pairs of opposite sides are congruent.
Example:
If AD = 3x + 6 and BC = x + 18, then x = _______ & AD =
________
A
B
C
D
E

6. PROOF
Statements Reasons
1. 𝐸𝑉 || 𝐿𝑂, 𝐸𝐿 || 𝑉𝑂 Given
1. 𝐿𝑉 is a diagonal
𝐸𝑂 is a diagonal
Definition of diagonal
1. ∠𝐿𝑂𝐸 ≅ ∠𝑉𝐸𝑂, ∠𝐿𝐸𝑂 ≅ ∠𝑉𝑂𝐸 Alternate Interior Angle are congruent
1. ∠𝐸𝐿𝑉 ≅ ∠𝑂𝑉𝐿, ∠𝐿𝑉𝐸 ≅ ∠𝑉𝐿𝑂 Alternate Interior Angle are congruent
1. 𝐸𝑂 ≅ 𝐸𝑂 , 𝐿𝑉 ≅ 𝐿𝑉 Reflexive
1. ⊿𝐿𝑂𝐸 ≅ ⊿𝑉𝐸𝑂 , ⊿𝐿𝑂𝑉 ≅ ⊿𝑉𝐸𝐿
ASA
1. ∠𝐸𝐿𝑂 ≅ ∠𝐸𝑉𝑂, ∠𝐿𝐸𝑉 ≅ ∠𝑉𝑂𝐿
Corresponding parts of congruent
triangles are congruent. (CPCTC)

7. Theorem 2: If a quadrilateral is a parallelogram, then
the two pairs of opposite angles are congruent.
Example:

8. PROOF
Statements Reasons
1. 𝐶𝐴 || 𝐸𝑅 Given
1. 𝐶𝐸 is a transversal
𝐴𝑅 is a transversal
Definition of a transversal
1. m∠𝐶 + m∠𝐸 = 180
m∠𝐴 + m∠𝑅 = 180
Angles on the same side of a
transversal are supplementary.
1. 𝐶𝐸 || 𝑅𝐴 Given
1. 𝐶𝐴 is a transversal
𝐴𝑅 is a transversal Definition of a transversal
1. m∠𝐶 + m∠𝐴 = 180 Angles on the same side of a

9. Theorem 3: If a quadrilateral is a parallelogram, then
the consecutive angles are supplementary.
Example:
Find the measure of angle Y.

10. PROOF
Statements Reasons
1.LOVE is a parallelogram Given
1.𝐿𝑂 ≅ 𝐸𝑉 Definition of a parallelogram
1.∠OLS ≅ ∠EVS
∠LOS ≅ ∠VES
Angles on the same side of a
transversal are
supplementary.
1.⊿LOS ≅ ⊿VES
ASA
Corresponding parts of

11. Theorem 4: If the diagonals of the quadrilateral bisect
each other, then the quadrilateral is a parallelogram.
Example:
Find the measure of segments KU and UM.
K
U
J
M
L

12. PROOF
Statements Reasons
1.ABCD is a parallelogram Given
1.𝐴𝐵 || 𝐶𝐷, 𝐴𝐷 || 𝐵𝐶
Definition of a
parallelogram
1.𝐴𝐶 is a diagonal Definition of diagonal
1.∠𝐵𝐴𝐶 ≅ ∠𝐷𝐶𝐴, ∠𝐷𝐴𝐶 ≅
∠𝐵𝐶𝐴
Alternate Interior
Angle are congruent
1.𝐴𝐶 ≅ 𝐴𝐶 Reflexive Property

13. A
D
B
C
Theorem 5: If a quadrilateral is a parallelogram, then
diagonals form two congruent triangles.
Example:
Given: ABCD is a Parallelogram
AC is a diagonal
Prove: ⊿ABC ≅⊿CDA