DWG NO. 06.3 — Lesson 3 of 5

Proving a Quadrilateral Is a Parallelogram

Unit 6: Quadrilaterals and Polygons · ~15–30 min

Objective

Determine whether a given quadrilateral must be a parallelogram, using one of five sufficient conditions.

Turning the properties around

Lesson 6.2 covered what's true if a shape is a parallelogram. Now flip the logic: given only a few measurements about a quadrilateral, when can you conclude it's a parallelogram in the first place? It turns out any one of the following conditions is enough on its own — you don't need all of them at once.

Both pairs of opposite sides parallel — this is the definition itself.
Both pairs of opposite sides congruent.
Both pairs of opposite angles congruent.
Diagonals that bisect each other.
One pair of opposite sides that is both parallel and congruent.

Each of these is a shortcut proven from the definition using triangle congruence, so any single one guarantees all the others follow. Mixing partial evidence — say, one pair of sides congruent but not the other — is not enough and doesn't guarantee a parallelogram.

If diagonals AC and BD bisect each other, ABCD is guaranteed to be a parallelogram — no angle or side info needed.

Worked Example 1 · Using congruent diagonals

ProblemQuadrilateral EFGH has diagonals that intersect at point M, with EM = 7, MG = 7, FM = 5, and MH = 5. Is EFGH a parallelogram?

1.EM = MG means diagonal EG is bisected at M. FM = MH means diagonal FH is also bisected at M.

2.Since both diagonals bisect each other, the Diagonals condition is satisfied.

Yes — EFGH is a parallelogram.

Worked Example 2 · Solving for x to guarantee a parallelogram

ProblemIn quadrilateral WXYZ, WX is parallel to ZY, WX = 3x + 2, and ZY = 5x − 8. Find x so that WXYZ is a parallelogram.

1.One pair of sides (WX, ZY) is already parallel, so making them congruent as well satisfies the "parallel and congruent" condition.

2.Set 3x + 2 = 5x − 8, so 10 = 2x, giving x = 5.

x = 5

Guided practice

1.Quadrilateral PQRS has PQ = RS = 12 and QR = SP = 8. Is PQRS necessarily a parallelogram? Which condition applies?
Show answer
Yes. Both pairs of opposite sides are congruent, which is one of the five sufficient conditions.
2.Quadrilateral ABCD has m∠A = m∠C = 70° and m∠B = m∠D = 110°. Is ABCD a parallelogram?
Show answer
Yes. Both pairs of opposite angles are congruent, satisfying the Opposite Angles condition.
3.Quadrilateral JKLM has JK = 9 and LM = 9, but only JK is parallel to LM (not confirmed for the other pair). Is that enough to prove it's a parallelogram?
Show answer
Yes — one pair of opposite sides that is both parallel and congruent is a sufficient condition on its own.
4.Diagonals of quadrilateral RSTU meet at point N, with RN = 6, NT = 6.2. Is RSTU guaranteed to be a parallelogram?
Show answer
Not from this alone. RN and NT aren't equal, so this diagonal isn't confirmed bisected, and no other condition has been shown — you'd need more information.
5.Explain why "one pair of opposite sides congruent, and the other pair parallel" is not on the list of sufficient conditions, even though it sounds similar to the valid ones.
Show answer
Mixing a congruence fact about one pair with a parallel fact about the other pair doesn't pin the shape down — you can construct a non-parallelogram trapezoid-like quadrilateral that satisfies that mixed condition, so it isn't guaranteed.

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