Properties of parallelograms

key notes :

Definition:

  • A parallelogram is a quadrilateral with two pairs of opposite sides parallel.

Properties of Parallelograms:

  • Opposite sides are equal: If a quadrilateral is a parallelogram, then its opposite sides are congruent.

AB = CD and BC = AD

  • Opposite angles are equal: The opposite angles in a parallelogram are congruent.

∠A = ∠C and ∠B = ∠D

  • Consecutive angles are supplementary: Any two adjacent angles add up to 180°.

∠A + ∠B = 180∘ , ∠B + ∠C = 180∘ , etc.

  • Diagonals bisect each other: The diagonals of a parallelogram cut each other into two equal halves.

AO = OC and BO = OD

  • Sum of interior angles: The total sum of all interior angles is 360°.

Special Types of Parallelograms:

  • Rectangle: A parallelogram with all angles 90°.
  • Rhombus: A parallelogram with all sides equal.
  • Square: A parallelogram that is both a rectangle and a rhombus (all sides equal, all angles 90°).

Conditions for a Quadrilateral to be a Parallelogram:

  • Both pairs of opposite sides are parallel.
  • Both pairs of opposite sides are equal.
  • One pair of opposite sides is both parallel and equal.
  • Diagonals bisect each other.
  • Opposite angles are equal.

Learn with an example

Find the value of c in parallelogram EFGH.

c =________

First, find the pair of opposite sides whose lengths are given in terms of c. The length of EH is 10c and the length of FG is c+9.

Since opposite sides of a parallelogram are congruent, set EH equal to FG and solve for c.

EH = FG

10c = c+9 Plug in EH=10c and FG=c+9

9c = 9 Subtract c from both side

c = 1 Divide both sides by 9

So, c=1.

Find the value of u in rhombus PQRS.

u=____°

First, find the two angles whose measures are in terms of u. The measure of ∠Q is u and the measure of ∠R is u+76°. Notice that ∠Q and ∠R are consecutive angles.

Consecutive angles in a rhombus are supplementary, so set the sum of ∠Q and ∠R equal to 180° and solve for u.

∠Q +∠R = 180°

u + (u + 76°) = 180° Plug in ∠Q = u and ∠R = u + 76°

2u + 76° = 180° Combine like terms

2u = 104° Subtract 76° from both sides

u = 52° Divide both sides by 2

So, u = 52°.

Find the value of t in rhombus PQRS.

t= _____°

First, find the two angles whose measures are in terms of t. The measure of ∠R is t and the measure of ∠S is t + 76°. Notice that ∠R and ∠S are consecutive angles.

Consecutive angles in a rhombus are supplementary, so set the sum of ∠R and ∠S equal to 180° and solve for t.

∠R + ∠S = 180°

t + (t + 76°) = 180° Plug in ∠R = t and ∠S = t + 76°

2t + 76° = 180° Combine like terms

2t = 104° Subtract 76° from both sides

t = 52° Divide both sides by 2

So, t = 52°.

Let’s practice!