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6.4 – Proving Quadrilaterals are Parallelograms
Theorems to Prove Quadrilaterals are Parallelograms:
 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is
a parallelogram.
 If an angle of a quadrilateral is supplementary to both consecutive angles, then the
quadrilateral is a parallelogram.
 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the
quadrilateral is a parallelogram.
Prove: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
B
A
̅̅̅̅ ≅ 𝐶𝐷
̅̅̅̅ and 𝐴𝐷
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅
Given: 𝐴𝐵
Prove: ABCD is a
Statement
Reason
D
C
Is there enough information to prove the quadrilateral is a parallelogram? If so, write a
theorem as justification why it is a parallelogram.
B
A
E
D
C
̅̅̅̅ and 𝐴𝐷
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅
3) ̅̅̅̅
𝐴𝐵||𝐶𝐷
̅̅̅̅ ||𝐵𝐶
̅̅̅̅ and 𝐴𝐷
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅
1) 𝐴𝐷
̅̅̅̅ and 𝐷𝐸
̅̅̅̅ ≅ ̅̅̅̅
2) ̅̅̅̅
𝐴𝐸 ≅ 𝐸𝐶
𝐸𝐵
4) ∠ADC ≅ ∠CBA and
5) ∠DAB is supplementary to ∠ADC
6) ΔAED ≅ ΔCEB
∠ABC is supplementary to ∠BCD
∠BAD ≅ ∠DCB
Determine the value of x and y so that the quadrilateral is a parallelogram.
1)
2)
3)
B
A
E
D
C
AE = x2 – 45, EC = -3x - 5
DE = 2y2, EB = -3y + 2
Proofs:
1) Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷, ̅̅̅̅
𝐴𝐹 ≅ ̅̅̅̅
𝐵𝐶 , ∠AFD ≅ ∠ADF
Prove: ABCD is a parallelogram
3) Given:
̅̅̅̅ ⊥ ̅̅̅̅̅
̅̅̅̅ ⊥ 𝑌𝑃
̅̅̅̅
WXYZ, 𝑍𝑂
𝑊𝑂 , 𝑋𝑃
Prove: WOYP is a parallelogram
2) Given:
WXYZ
Prove: ΔWOX ≅ ΔYOZ using AAS
̅̅̅̅̅
4) Given: ΔRQP ≅ ΔONP, R is midpoint of 𝑀𝑄
Prove: MRON is a parallelogram
(Give a reason for each step)
5) Given:
̅̅̅̅ ≅ 𝑈𝑅
̅̅̅̅
PQRS, 𝑃𝑇
̅̅̅̅ ≅ 𝑆𝑈
̅̅̅̅
Prove: 𝑄𝑇
Q
P
T
U
S
R
Using Coordinate Geometry
Show that the coordinates A(2, -1), B(1, 3), C(6, 5), and D(7, 1) are vertices of a parallelogram.
What are the 5 ways we can prove a quadrilateral is a parallelogram?
1)
2)
3)
4)
5)
Which of these are we able to show with Algebra /formulas using the given vertices?
Use the space below to show the work to justify these ways.