Proving a Quadrilateral Is a Parallelogram

Knowing the properties of parallelograms lets you find missing measures when a figure is already labeled a parallelogram. This skill runs the other direction: you are handed a plain quadrilateral with some given information, and you have to decide whether that information guarantees it is a parallelogram.

There are five conditions — five tests — and any single one of them is enough. The real skill is knowing which given facts pass a test and which almost-right combinations do not.

The five tests

A quadrilateral is a parallelogram if any one of these is true: both pairs of opposite sides are parallel (this is the definition), both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, the diagonals bisect each other, or one pair of opposite sides is both parallel and congruent.

Notice what the last test asks for: the same pair of sides must be both parallel and congruent. That double condition on a single pair is what makes it strong enough — and it is where the traps live.

The combinations that fail

One pair of sides parallel and the other pair congruent does not work. A quadrilateral with $\overline{AB} \parallel \overline{CD}$ and $AD = BC$ could be a parallelogram — but it could also be an isosceles trapezoid, which has exactly one parallel pair and congruent legs. Since the given facts allow a non-parallelogram, they prove nothing.

Congruent diagonals also fail. An isosceles trapezoid has congruent diagonals too, so $AC = BD$ on its own does not force a parallelogram. What the diagonals must do is bisect each other — each one passes through the midpoint of the other.

The quick check for any proposed test: ask whether an isosceles trapezoid could satisfy it. If yes, the test is not valid.

The figure is that trap: $\overline{AB} \parallel \overline{DC}$ (one parallel pair) and congruent legs $\overline{AD}$ and $\overline{BC}$ — yet it is a trapezoid, not a parallelogram.

Reading the given information

In a marked diagram, tick marks show congruent segments, arrowheads on sides show parallel pairs, and matching angle arcs show congruent angles. List exactly what the marks guarantee — nothing more — and then check that list against the five tests. If the marks show both pairs of opposite sides congruent, that is a valid test. If they show one parallel pair and a congruent pair of legs, it is not.

In diagonal problems, the given lengths are usually the four pieces made by the intersection point. To pass the bisecting test, the two pieces of each diagonal must match: $AE = EC$ and $BE = ED$ .

Worked examples

Example 1: diagonals that bisect each other

The diagonals of quadrilateral $PQRS$ intersect at $M$ , with $PM = 6$ , $MR = 6$ , and $QM = 4$ . What must $MS$ equal to guarantee $PQRS$ is a parallelogram?

The test requires each diagonal to cut the other in half

PM = MR \text{ and } QM = MS

The first diagonal already checks out

PM = MR = 6

Match the second pair

MS = QM = 4

Answer: $MS = 4$

Example 2: one pair, parallel and congruent

In quadrilateral $ABCD$ , $\overline{AB} \parallel \overline{DC}$ and $AB = DC = 12$ . Is $ABCD$ a parallelogram?

Identify what is given about the pair

\overline{AB}

and

\overline{DC}

\overline{AB} \parallel \overline{DC} \text{ and } AB = DC

The same pair of opposite sides is both parallel and congruent — that is one of the five tests

Conclude: the test passes, so

ABCD

must be a parallelogram

Answer: Yes — one pair of opposite sides is both parallel and congruent.

Example 3: a combination that proves nothing

In quadrilateral $PQRS$ , $\overline{PQ} \parallel \overline{SR}$ and $PS = QR$ . Must $PQRS$ be a parallelogram?

Compare against the tests: the parallel pair is

\overline{PQ}

and

\overline{SR}

, but the congruent pair is different sides

PS = QR

An isosceles trapezoid also has one parallel pair and congruent legs, so these facts fit a non-parallelogram

Conclude: the given information is not enough

Answer: No — $PQRS$ could be an isosceles trapezoid.

Try one yourself

Common questions

Do I need to verify all five conditions?

No. Any one of the five tests, fully satisfied, is enough on its own. The tests are five different doors into the same room.

Why aren't congruent diagonals enough?

Because an isosceles trapezoid also has congruent diagonals, and it is not a parallelogram. Congruent tells you the diagonals have equal lengths; the parallelogram test needs them to bisect each other, which is about where they cross, not how long they are.

Is one pair of parallel sides enough?

No — that only makes the figure a trapezoid. To use a single pair of sides, that same pair must be both parallel and congruent.

Which test is easiest on the coordinate plane?

The diagonals test. Compute the midpoint of each diagonal with $\left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)$ ; if both diagonals have the same midpoint, they bisect each other and the quadrilateral is a parallelogram. It is one calculation instead of four slope or distance checks.

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Proving a Quadrilateral Is a Parallelogram

The five tests

The combinations that fail

Reading the given information

Worked examples

Example 1: diagonals that bisect each other

Example 2: one pair, parallel and congruent

Example 3: a combination that proves nothing

Try one yourself

Common questions

Keep learning

Properties of Parallelograms

Polygon Interior & Exterior Angle Sums

Rectangles, Rhombi & Squares

Trapezoids & Kites