Properties of Parallelograms

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A parallelogram is a quadrilateral with both pairs of opposite sides parallel. In every parallelogram, opposite sides are congruent, opposite angles are congruent, consecutive angles add to 180°, and the diagonals bisect each other. These four properties let you find missing sides, angles and diagonal lengths.

Before this lesson: You should know Interior and Exterior Angles of Polygons, Parallel Lines and Transversals and CPCTC.

What Makes a Parallelogram

Think of a rectangle that has been pushed sideways, like a cardboard box squashed at an angle. The top and bottom stay parallel, and so do the two sides. That shape is a parallelogram. Rectangles, rhombuses and squares are special parallelograms.

A quadrilateral’s angles add to 360°. In a parallelogram, the angles come in two matching pairs, and any two angles next to each other add to 180°. Angles next to each other are called consecutive angles.

Parallelogram ABCD with A at the bottom left, B at the bottom right, C at the top right and D at the top left. Arrow marks show AB parallel to DC and AD parallel to BC. AB and DC have one tick mark, and AD and BC have two tick marks. Angles A and C have one orange arc, and angles B and D have two blue arcs. The diagonals AC and BD cross at E, and each diagonal has matching marks on its two halves, showing that E is the midpoint of both.
Matching marks show the four properties: congruent opposite sides, congruent opposite angles, and diagonals that cut each other in half at E.

Properties of Parallelograms

In parallelogram ABCD with diagonals meeting at E:

Property In symbols
Opposite sides are parallel \overline{AB} \parallel \overline{DC} and \overline{AD} \parallel \overline{BC}
Opposite sides are congruent AB = DC and AD = BC
Opposite angles are congruent m\angle A = m\angle C and m\angle B = m\angle D
Consecutive angles are supplementary m\angle A + m\angle B = 180^\circ
The diagonals bisect each other AE = EC and BE = ED

Why are consecutive angles supplementary? Sides AD and BC are parallel, and AB is a transversal. Angles A and B are same-side interior angles, so they add to 180°.

How to Use Parallelogram Properties

  1. Label the parallelogram in order around the shape, like ABCD.
  2. For sides, set opposite sides equal.
  3. For angles, set opposite angles equal, or make consecutive angles add to 180°.
  4. For diagonals, set the two halves of each diagonal equal. The full diagonal is twice one half.
  5. Solve, then substitute back to find the measures.

Worked Examples

Example 1: Word problem with a garden bed

A garden bed is shaped like a parallelogram ABCD. Side AB is 12 ft, side BC is 7 ft, and the angle at A is 65°. Find the other sides, the other angles and the perimeter.

  1. Opposite sides are congruent: CD = AB = 12 ft and AD = BC = 7 ft.
  2. Opposite angles are congruent: m\angle C = m\angle A = 65^\circ.
  3. Consecutive angles add to 180°: m\angle B = 180^\circ - 65^\circ = 115^\circ.
  4. Angle D is opposite angle B: m\angle D = 115^\circ.
  5. Add the four sides: 12 + 7 + 12 + 7 = 38
  6. Check the angles: 65 + 115 + 65 + 115 = 360

Answer: CD = 12 ft and AD = 7 ft. The angles are 65°, 115°, 65° and 115°. The perimeter is 38 ft.

Example 2: Solve for x with opposite sides

In parallelogram ABCD, AB = 3x + 2 and CD = 5x - 10. Find x and AB.

  1. Opposite sides are congruent: 3x + 2 = 5x - 10
  2. Subtract 3x from both sides (see equations with variables on both sides): 2 = 2x - 10
  3. Add 10 to both sides: 12 = 2x
  4. Divide both sides by 2: 6 = x
  5. Find AB: 3(6) + 2 = 18 + 2 = 20
  6. Check with CD: 5(6) - 10 = 30 - 10 = 20

Answer: x = 6 and AB = 20.

Example 3: Solve for y with consecutive angles

In parallelogram ABCD, m\angle A = (2y + 10)^\circ and m\angle B = (3y + 20)^\circ. Find y and each angle of the parallelogram.

  1. Angles A and B are consecutive, so they add to 180°: (2y + 10) + (3y + 20) = 180
  2. Combine like terms (see combining like terms): 5y + 30 = 180
  3. Subtract 30 from both sides: 5y = 150
  4. Divide both sides by 5: y = 30
  5. Find angle A: 2(30) + 10 = 60 + 10 = 70
  6. Find angle B: 3(30) + 20 = 90 + 20 = 110
  7. Opposite angles match: m\angle C = 70^\circ and m\angle D = 110^\circ.

Answer: y = 30. The angles are 70°, 110°, 70° and 110°.

Example 4: Diagonals that bisect each other

The diagonals of parallelogram ABCD meet at E. AE = 2z + 1 and EC = 4z - 9. Also BE = 7. Find z, AC and BD.

  1. The diagonals bisect each other, so AE = EC: 2z + 1 = 4z - 9
  2. Subtract 2z from both sides: 1 = 2z - 9
  3. Add 9 to both sides: 10 = 2z
  4. Divide both sides by 2: 5 = z
  5. Find AE: 2(5) + 1 = 10 + 1 = 11
  6. AC is twice AE: AC = 2 \times 11 = 22
  7. BD is twice BE: BD = 2 \times 7 = 14

Answer: z = 5, AC = 22 and BD = 14.

Example 5: Prove that opposite sides are congruent

Given: ABCD is a parallelogram. Draw diagonal AC.
Prove: \overline{AB} \cong \overline{CD} and \overline{BC} \cong \overline{DA}

Statements Reasons
1. ABCD is a parallelogram Given
2. \overline{AB} \parallel \overline{DC} and \overline{AD} \parallel \overline{BC} Definition of a parallelogram
3. \angle BAC \cong \angle DCA Alternate interior angles theorem (AB ∥ DC)
4. \angle BCA \cong \angle DAC Alternate interior angles theorem (AD ∥ BC)
5. \overline{AC} \cong \overline{AC} Reflexive property of congruence
6. \triangle ABC \cong \triangle CDA ASA (steps 3, 5 and 4)
7. \overline{AB} \cong \overline{CD} and \overline{BC} \cong \overline{DA} CPCTC

The same pair of triangles also gives \angle B \cong \angle D by CPCTC. That is why opposite angles are congruent too.

Common Mistakes

Mistake Example of the error Fix
Setting consecutive angles equal 2y + 10 = 3y + 20 for angles A and B Consecutive angles add to 180°. Only opposite angles are equal.
Setting next-door sides equal AB = BC in any parallelogram Only opposite sides are congruent. Adjacent sides match only in a rhombus.
Thinking the diagonals are congruent AC = BD in every parallelogram The diagonals bisect each other, but they are equal only in a rectangle.
Using half a diagonal as the whole AE = 11, so “AC = 11” AE is half of AC, so AC = 22.
Labeling vertices out of order Calling a parallelogram ACBD Name the vertices in order around the shape, so opposite sides are AB and CD.

Parallelograms on the Coordinate Plane

Practice Problems

  1. In parallelogram ABCD, AB = 15. Find CD.
  2. In parallelogram ABCD, m\angle B = 108^\circ. Find m\angle D and m\angle A.
  3. In parallelogram ABCD, AD = 4x - 3 and BC = 2x + 7. Find x and AD.
  4. In parallelogram ABCD, m\angle A = (5x)^\circ and m\angle C = (3x + 40)^\circ. Find x and m\angle A.
  5. In parallelogram ABCD, m\angle A = (4y - 6)^\circ and m\angle D = (2y + 18)^\circ. Find y and both angles.
  6. The diagonals of parallelogram ABCD meet at E. AE = 9. Find AC. Also, BE = x + 3 and ED = 2x - 5. Find x and BD.
  7. Parallelogram ABCD has vertices A(1, 1), B(6, 1), C(8, 4) and D(3, 4). Show that its diagonals bisect each other.
  8. A parallelogram has sides of 9 cm and 14 cm. Find its perimeter.
  9. Complete the proof. Given: ABCD is a parallelogram with diagonals meeting at E. Prove: the diagonals bisect each other. Statements: (1) ABCD is a parallelogram. (2) \overline{AB} \parallel \overline{DC}. (3) \angle ABE \cong \angle CDE. (4) \angle BAE \cong \angle DCE. (5) \overline{AB} \cong \overline{CD}. (6) \triangle ABE \cong \triangle CDE. (7) \overline{AE} \cong \overline{CE} and \overline{BE} \cong \overline{DE}. (8) The diagonals bisect each other. Give a reason for each statement.
  10. Can a parallelogram have two consecutive angles of 70° and 120°? Explain.
Show answers
  1. Opposite sides are congruent
    CD = 15
  2. Opposite angles are congruent: m\angle D = 108^\circ
    Consecutive angles add to 180: 180 - 108 = 72, so m\angle A = 72^\circ
  3. Opposite sides are equal: 4x - 3 = 2x + 7
    Subtract 2x: 2x - 3 = 7
    Add 3: 2x = 10
    Divide by 2: x = 5
    Find AD: 4(5) - 3 = 20 - 3 = 17
  4. Opposite angles are equal: 5x = 3x + 40
    Subtract 3x: 2x = 40
    Divide by 2: x = 20
    Find the angle: 5(20) = 100, so m\angle A = 100^\circ
  5. Consecutive angles add to 180: (4y - 6) + (2y + 18) = 180
    Combine like terms: 6y + 12 = 180
    Subtract 12: 6y = 168
    Divide by 6: y = 28
    Angle A: 4(28) - 6 = 112 - 6 = 106, so 106°
    Angle D: 2(28) + 18 = 56 + 18 = 74, so 74°
  6. AC is twice AE: 2 \times 9 = 18
    The halves of BD are equal: x + 3 = 2x - 5
    Subtract x: 3 = x - 5
    Add 5: x = 8
    BE: 8 + 3 = 11
    BD: 2 \times 11 = 22
  7. Midpoint of AC: \left(\tfrac{1 + 8}{2}, \tfrac{1 + 4}{2}\right) = (4.5, 2.5)
    Midpoint of BD: \left(\tfrac{6 + 3}{2}, \tfrac{1 + 4}{2}\right) = (4.5, 2.5)
    The midpoints are the same point, so the diagonals bisect each other.
  8. Opposite sides are equal, so add 9 + 14 + 9 + 14 = 46
    The perimeter is 46 cm.
  9. (1) Given
    (2) Definition of a parallelogram
    (3) Alternate interior angles theorem
    (4) Alternate interior angles theorem
    (5) Opposite sides of a parallelogram are congruent
    (6) ASA (steps 4, 5 and 3)
    (7) CPCTC
    (8) Definition of segment bisector (E is the midpoint of both diagonals)
  10. No. Consecutive angles of a parallelogram add to 180°, but 70 + 120 = 190.

FAQ

What are the properties of a parallelogram?

Opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and the diagonals bisect each other.

Do the diagonals of a parallelogram bisect the angles?

Not usually. They bisect each other, which means they cut each other in half. They bisect the angles only in a rhombus or a square.

Are the diagonals of a parallelogram equal?

Only when the parallelogram is a rectangle. In other parallelograms, one diagonal is longer than the other.

What do the angles of a parallelogram add up to?

All four angles add to 360°, like any quadrilateral. Any two consecutive angles add to 180°.

Summary

  • A parallelogram has two pairs of parallel opposite sides.
  • Opposite sides are congruent.
  • Opposite angles are congruent, and consecutive angles add to 180°.
  • The diagonals bisect each other, so each half of a diagonal is equal.
  • The properties come from congruent triangles made by a diagonal.

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Next lesson: Proving a Quadrilateral Is a Parallelogram

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