Properties of Quadrilaterals

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A quadrilateral is any four-sided figure.

In any quadrilateral, the sum of all interior angles is 360ᵒ.

 

Quadrilateral properties are discussed below, which are used to logically test whether a certain shape is one kind of quadrilateral.

 

Parallelogram

A parallelogram is a quadrilateral with parallel opposite sides.

Properties of a parallelogram:

  1. Opposite sides are equal
  2. Opposite angles are equal
  3. Diagonals bisect each other
  4. Each diagonal bisects the parallelogram into two congruent triangles

(Bisect means cut exactly in two halves)

 

Rectangle

A rectangle is a parallelogram with all angles forming a 90ᵒ angle.

Properties of a rectangle:

  1. Opposite sides are equal
  2. Opposite angles are equal
  3. Diagonals bisect each other
  4. Each diagonal bisects the parallelogram into two congruent triangles
  5. Diagonals are equal

 

Rhombus

A rhombus is a parallelogram with a pair of equal adjacent sides.

Properties of a rhombus:

  1. Opposite sides are equal
  2. Opposite angles are equal
  3. Diagonals bisect each other
  4. Each diagonal bisects the parallelogram into two congruent triangles
  5. Diagonals bisect at right angles
  6. Diagonals bisect the angles of the rhombus

 

Square

A square is a rectangle with a pair of equal adjacent sides.

  1. Opposite sides are equal
  2. Opposite angles are equal
  3. Diagonals bisect each other
  4. Each diagonal bisects the parallelogram into two congruent triangles
  5. Diagonals are equal
  6. Diagonals are perpendicular
  7. Diagonals make angles of 45° with the sides

 

Trapezium

A trapezium is a quadrilateral with one pair of parallel sides.

 

Kite

A kite is a quadrilateral with two pairs of adjacent equal sides.

Properties of a kite:

  1. Two adjacent sides are equal
  2. Diagonals bisect at 90° angles
  3. One diagonal bisects the kite to two congruent triangles
  4. One pair opposite of angles is equal

Examples

 

1. ABCD is a square. Lines are drawn from C to M and N, the midpoints AD and AB respectively. Prove that MC = NC.

Show Answer

Using similarity of triangles, we check if the triangles △CDM and △CBN are congruent.

 

Since MD and NB are half of a square sides, therefore MD = NB. (S)

∠ADC and ∠ABC are equal right angles property of a square. (A)

While, DC = BC because a square has equal sides. (S)

Therefore, by SAS, the two triangles △CDM and △CBN are congruent.

Hence, the sides MC and NC are equal because they are corresponding sides of congruent triangles.

 

 

2. If A = (1, 5), B = (4, 2), and C = (2, -3), find the coordinates of D such that ABCD is a parallelogram.

Show Answer

 

The question is drawn on the above graph, and graphically we could understand where the points are.

A parallelogram has 2 pairs of parallel sides, and therefore the distance BA must be equal to the distance CD. Therefore,

 

From point B to point A, the distance moves 3 to the left and then 3 upwards. The same could be done from point C to D to complete the parallelogram, which we found to be the point D(-1, 0). This method complies with the properties of parallelograms, with 2 pairs of parallelograms and equal opposite angles.