Proving a Quadrilateral is a Parallelogram
In 9.2.1, we assumed a parallelogram and found its properties. Now we reverse it: given a quadrilateral, how do you proveit's a parallelogram?
Introduction
There are five sufficient conditions— prove any ONE of them and you've proven the quadrilateral is a parallelogram.
Past Knowledge
Parallelogram properties (9.2.1). Congruence proofs (5.2). Parallel lines (3.1).
Today's Goal
Learn and apply the 5 ways to prove a quadrilateral is a parallelogram.
Future Success
Special parallelograms (9.2.3–9.2.5). Coordinate proofs (9.3.4).
Key Concepts
5 Ways to Prove a Parallelogram
- Method 1: Show both pairs of opposite sides are parallel
- Method 2: Show both pairs of opposite sides are congruent
- Method 3: Show both pairs of opposite angles are congruent
- Method 4: Show the diagonals bisect each other
- Method 5: Show one pair of sides is both parallel AND congruent
⚠️ NOT Sufficient
- One pair of sides congruent, a different pair parallel → not enough
- One pair of sides parallel only → not enough
- One pair of angles equal only → not enough
Theorem & Proof
Two-Column Proof: Method 5 (One Pair Parallel + Congruent)
Given: Quadrilateral with and
Prove: is a parallelogram
| # | Statement | Reason |
|---|---|---|
| 1 | Draw diagonal | Two points determine a line |
| 2 | Alt. interior angles () | |
| 3 | Reflexive Property | |
| 4 | SAS Congruence (, step 2, step 3) | |
| 5 | CPCTC | |
| 6 | → is a parallelogram | Converse of Alt. Int. Angles; both pairs of opp. sides parallel |
∎ One pair of sides that is both parallel and congruent guarantees the other pair is also parallel.
Worked Examples
Method 2 — Opposite Sides Congruent
WXYZ has WX = YZ = 8 and WZ = XY = 12. Is it a parallelogram?
Both pairs of opposite sides are congruent → YES, parallelogram (Method 2).
Yes — both pairs of opposite sides are congruent.
Method 4 — Diagonal Bisection
Diagonals of PQRS meet at M. PM = 5, MR = 5, QM = 8, MS = 8. Is it a parallelogram?
PM = MR and QM = MS → diagonals bisect each other → YES (Method 4).
Yes — diagonals bisect each other.
Not Enough Info
ABCD has AB ∥ DC and AD = BC. Is it a parallelogram?
AB ∥ DC (one pair parallel) and AD = BC (one pair congruent) — but these are different pairs!
This could be a parallelogram OR an isosceles trapezoid. Not enough info.
Cannot determine — the parallel pair and congruent pair must be the same pair (Method 5) or both pairs (Methods 1, 2).
Common Pitfalls
Mixing Up Pairs
Method 5 requires the SAME pair to be both parallel and congruent. One pair parallel + a different pair congruent could be a trapezoid.
One Pair Only
Showing only one pair of sides parallel (without congruence) is not enough. That's a trapezoid's definition, not a parallelogram's.
Real-Life Applications
Furniture Making — Checking Square Frames
To verify a door frame is a parallelogram, a carpenter measures both diagonals. If they bisect each other, the frame is guaranteed to be a parallelogram (Method 4).
Bridge Trusses
Engineers verify that structural panels are parallelograms by checking that opposite members are both the same length and parallel (Method 5).
Practice Quiz
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