If the opposite sides of a quadrilateral are equal, is it a parallelogram?

Yes, for a convex quadrilateral: a diagonal cuts it into two triangles congruent by SSS, their equal angles at the diagonal are alternate angles, and equal alternate angles make both pairs of opposite sides parallel.

Why do equal alternate angles make lines parallel?

If the two lines met, they would form a triangle with the transversal, and one of the alternate angles would be an exterior angle of that triangle equal to an interior opposite angle. The exterior angle theorem says an exterior angle is always larger, so the lines cannot meet.

What are the ways to show a quadrilateral is a parallelogram?

Both pairs of opposite sides parallel (the definition), both pairs equal, both pairs of opposite angles equal, the diagonals bisecting each other, or one pair of opposite sides both equal and parallel.

Equal Opposite Sides Make a Parallelogram

How to use

▶ Play runs the proof, Next → and ← Back step one scene at a time, Reset restores the opening scene and the original figure. Learn more about the controls
Drag $A$ , $B$ or $D$ ( $C$ follows); the readout under the canvas shows the two pairs of alternate angles at the diagonal. Learn more about why it holds
Labels toggles the point names, Measures writes each marked angle's size beside its arc. The Step explanations log keeps every step so far. Learn more about reading a scene

Proof?AB = CD and AD = CB. The diagonal BD cuts the quadrilateral into the triangles ABD and CDB, which have three pairs of equal sides, so they are congruent (SSS). Their equal angles at B and D are alternate angles for AB and DC, and for AD and BC, with the transversal BD; equal alternate angles make the lines parallel, so ABCD is a parallelogram. Drag A, B or D.Parallelogram from equal sides

DisplayLabelsMeasures

Drag A, B or D (C keeps the sides equal)

∠ABD = 51°,  ∠CDB = 51°,  ∠ADB = 66°,  ∠CBD = 66°

Step 1 / 6

Step explanations

1A quadrilateral with equal opposite sides

Given
ABCD is a convex quadrilateral with AB = CD (one tick) and AD = CB (two ticks). To prove: ABCD is a parallelogram, that is, AB ∥ DC and AD ∥ BC.

Drag A, B or D: C keeps the opposite sides equal. Learn more about the statement

Getting Started with the Visualizer

Equal Alternate Angles Give Parallels

Why It Holds

Dragging the Figure

Consequences

Getting Started with the Visualizer

DemoA run through the proof player

Step 0 of 7

Step through the proof, then drag

A

B

D

(

C

follows so that the opposite sides stay equal) and watch the two pairs of alternate angles in the readout stay equal. The picture below is the player with its parts numbered; the other tabs list what can be dragged, what each control does and what the step log shows.

Opening scene: the controls

Proof?Parallelogram from equal sidesDisplayLabelsMeasuresDrag A, B or D (C keeps the sides equal)

∠ABD = 51°,  ∠CDB = 51°,  ∠ADB = 66°,  ∠CBD = 66°

← Back▶ PlayNext →ResetNormal ▾Step 1 / 6

Step explanations

1A quadrilateral with equal opposite sides

Given
ABCD is a convex quadrilateral with AB = CD (one tick) and AD = CB (two ticks). To prove: ABCD is a parallelogram, that is, AB ∥ DC and AD ∥ BC.

1234567

1Proof — the theorem loaded in the player; the ? gives its idea in one line

2Display — Labels shows the point names, Measures writes each marked angle's size beside its arc

3Draggable points — A, B and D move freely; C follows so that CD = AB and CB = AD

4Readout — the line under the figure: the two pairs of alternate angles at the diagonal, live while you drag

5Back, Play, Next, Reset — step through the proof or run it; Reset returns to step 1 and restores the original figure

6Speed and step counter — how fast Play advances, and which of the 6 steps is on screen

7Step explanations — every step reached so far, with its justification tag and a link to the section that explains it; click an entry to jump back to it

Proof complete: step 6 of 6

Proof?Parallelogram from equal sidesDisplayLabelsMeasuresDrag A, B or D (C keeps the sides equal)

∠ABD = 51°,  ∠CDB = 51°,  ∠ADB = 66°,  ∠CBD = 66°

← Back↻ ReplayNext →ResetNormal ▾Step 6 / 6

Step explanations

1A quadrilateral with equal opposite sides

Given
ABCD is a convex quadrilateral with AB = CD (one tick) and AD = CB (two ticks). To prove: ABCD is a parallelogram, that is, AB ∥ DC and AD ∥ BC.

2The diagonal BD

Construction
Draw the diagonal BD. It cuts the quadrilateral into the triangles ABD and CDB, which share the side BD.

3ABD ≅ CDB

SSS
The two triangles have AB = CD, AD = CB, and the common side BD = DB: three pairs of equal sides. By SSS they are congruent, with A ↔ C, B ↔ D, D ↔ B.

4∠ABD = ∠CDB, ∠ADB = ∠CBD

Corresponding parts
Corresponding angles of the congruent triangles are equal: ∠ABD = ∠CDB = α (the angles opposite the equal sides AD and CB) and ∠ADB = ∠CBD = β (opposite AB and CD).

5AB ∥ DC and AD ∥ BC

Equal alternate angles
The angles α at B and at D lie on opposite sides of the transversal BD, between the lines AB and DC: they are alternate angles, and they are equal. Equal alternate angles make the lines parallel: if AB and DC met on one side, the triangle they formed with BD would have an exterior angle equal to an interior opposite angle, which the exterior angle theorem forbids. So AB ∥ DC; and in the same way the equal angles β make AD ∥ BC.

6ABCD is a parallelogram

Definition
Both pairs of opposite sides are parallel, which is the definition of a parallelogram: ABCD is a parallelogram. Together with the theorem that a parallelogram has equal opposite sides, this says that a quadrilateral is a parallelogram exactly when its opposite sides are equal. ∎

12345

1The two triangles — ABD and CDB, shaded on either side of the diagonal BD

2The alternate angles — α at B and at D, β at D and at B, matched across the diagonal

3The chevrons — one pair on AB and DC, two on AD and BC: the parallels found

4Justification tags — Given, Construction, SSS, Corresponding parts, Equal alternate angles, Definition

5Replay — at the last step Play becomes Replay and Next is off

The Statement

The parallelogram sides page proves that a parallelogram has equal opposite sides. This page proves the converse: if a convex quadrilateral

ABCD

has

AB = CD, \qquad AD = CB

then it is a parallelogram, that is,

AB \parallel DC

and

AD \parallel BC

. Together the two statements say that "opposite sides equal" is a test for a parallelogram, the one most often used in practice: measure the four sides and you know the shape.

The player keeps the hypothesis by construction:

C

is placed so that

CD = AB

and

CB = AD

(the only convex quadrilateral with those four sides in that order). Drag

A

B

D

; the readout shows the two pairs of alternate angles at the diagonal.

Opening scene, frozen

AB = CD (one tick) and AD = CB (two ticks).

The Diagonal

Draw the diagonal

BD

. It cuts the quadrilateral into two triangles,

ABD

on one side and

CDB

on the other, and both of them contain the diagonal as a side. The whole proof is about these two triangles.

The diagonal, frozen

BD drawn; the triangles ABD and CDB shaded.

Three Equal Sides

Compare the triangles

ABD

and

CDB

side by side:

AB = CD

and

AD = CB

are given, and

BD

is a side of both. Three pairs of equal sides, so by SSS congruence

\triangle ABD \cong \triangle CDB

with the vertices matched

A \leftrightarrow C

B \leftrightarrow D

D \leftrightarrow B

Three equal sides, frozen

AB = CD, AD = CB, BD common: ABD ≅ CDB by SSS.

The Alternate Angles

Congruent triangles have equal corresponding angles. The angle at

B

in the first triangle,

\angle ABD

, corresponds to the angle at

D

in the second,

\angle CDB

; both are opposite the equal sides

AD

and

CB

. Likewise

\angle ADB

corresponds to

\angle CBD

\angle ABD = \angle CDB = \alpha, \qquad \angle ADB = \angle CBD = \beta

Each pair sits on opposite sides of the diagonal

BD

, between two of the sides of the quadrilateral: they are alternate angles for the transversal

BD

The alternate angles, frozen

α at B and D, β at D and B: two pairs of equal alternate angles.

Equal Alternate Angles Give Parallels

The angles

\alpha

B

and at

D

are alternate angles between the lines

AB

and

DC

cut by

BD

, and they are equal. Equal alternate angles force the lines to be parallel (Euclid I.27): if

AB

and

DC

met at some point

X

on one side of

BD

, the triangle

XBD

would have at one of

B

D

an exterior angle equal to the interior angle at the other, and the exterior angle theorem says an exterior angle is always larger. So

AB \parallel DC

and the equal angles

\beta

give

AD \parallel BC

in the same way. This is the converse direction of the parallel lines page, which goes from parallels to equal angles.

Equal alternate angles give parallels, frozen

Chevrons on AB and DC, and on AD and BC.

Why It Holds

Both pairs of opposite sides are parallel, and that is what a parallelogram is:

ABCD \text{ is a parallelogram}

The end-of-proof mark ∎ in the log closes the argument. The tags name the facts used: the given figure, a construction, SSS, corresponding parts, equal alternate angles, and the definition.

One diagonal, one congruence, two pairs of alternate angles: the same diagonal that proves a parallelogram has equal opposite sides proves the converse, with SSS in place of ASA.

The conclusion, frozen

Both pairs of opposite sides parallel: a parallelogram.

Dragging the Figure

Drag

A

B

D

: the quadrilateral changes shape,

C

moves so that

CD = AB

and

CB = AD

stay true, the diagonal and the four marked angles follow, and the readout shows the two pairs of alternate angles staying equal. The figure stays a parallelogram throughout, as the theorem says it must.

The player keeps the quadrilateral convex and its sides long enough for the tick marks, and refuses a drag that would carry a vertex out of the frame or flatten the figure.

After dragging D, frozen

D moved up and to the right: a more slanted figure, still a parallelogram.

Consequences

• Tests for a parallelogram. Equal opposite sides, equal opposite angles, diagonals that bisect each other, or one pair of sides both equal and parallel: each is enough, and each has a proof of this shape.
• Rhombus. Four equal sides are in particular two pairs of equal opposite sides, so every rhombus is a parallelogram; the rhombus diagonals page starts from this.
• Constructing parallels. To draw through a point

P

a parallel to a line

AB

with compasses: mark

C

with

PC = AB

and

BC = AP

on the right side, and

PC \parallel AB

by this theorem.
• Midsegments. In the midsegment theorem a parallelogram is recognised from two equal and parallel sides, the sister test of this one.
• Vectors. The theorem is why two equal and parallel arrows

\overrightarrow{AB}

and

\overrightarrow{DC}

represent the same vector: their endpoints form a parallelogram.