Facts that Matter: Quadrilaterals

Table of Contents
1. Quadrilateral
2. Angle Sum Property of a Quadrilateral
3. Types of Quadrilaterals
4. Properties of Parallelograms
5. The Mid-Point Theorem
6. Converse of the Mid-Point Theorem
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1. A plane figure bounded by four sides is called a quadrilateral.
2. The sum of all the interior angles of a quadrilateral is 360º.
3. A quadrilateral is a parallelogramif
   • opposite sides are equal,
   • opposite angles are equal,
   • diagonals bisect each other,
   • a pair of opposite sides is equal and parallel.
4. A diagonal of a parallelogram divides it into two congruent triangles.
5. Diagonals of a rhombus bisect each other at right angles, and the converse is also true.
6. Diagonals of a square bisect each other at right angles and are equal, and the converse is also true.
7. Diagonals of a rectangle bisect each other, and the converse is also true.
8. A line segment joining the mid-points of any two sides of a triangle is parallel to the third side.
9. A line segment joining the mid-points of any two sides of a triangle is half of the third side.
10. The quadrilateral formed by joining the mid-points of the sides of any quadrilateral is a parallelogram.

Quadrilateral

A quadrilateral is a plane figure bounded by four line segments. If the vertices are A, B, C and D, the quadrilateral is denoted by ABCD. The sides are the line segments AB, BC, CD and DA. The diagonals are AC and BD.

• The points A, B, C and D are the vertices of quadrilateral ABCD.
• The line segments AB, BC, CD and DA are the sides of quadrilateral ABCD.
• The line segments AC and BD are called the diagonals of quadrilateral ABCD.

Note:

• Two sides having a common end point are called adjacent sides.
• Two sides having no common end point are called opposite sides.
• Two angles of a quadrilateral having a common arm are called consecutive angles.
• Two angles of a quadrilateral having no common arm are called opposite angles.

Angle Sum Property of a Quadrilateral

Statement: The sum of all four interior angles of a quadrilateral is 360º.

Proof:

Consider quadrilateral ABCD. Join diagonal BD to divide the quadrilateral into triangles ABD and BCD.

In triangle ABD, the sum of the interior angles is 180º.
∠1 + ∠A + ∠2 = 180º

In triangle BCD, the sum of the interior angles is 180º.
∠3 + ∠C + ∠4 = 180º

Adding the two equations, we get
(∠1 + ∠A + ∠2) + (∠3 + ∠C + ∠4) = 180º + 180º
(∠1 + ∠3) + ∠A + ∠C + (∠2 + ∠4) = 360º

Since ∠1 + ∠3 = ∠B and ∠2 + ∠4 = ∠D, we obtain
∠A + ∠B + ∠C + ∠D = 360º

Thus, the sum of the four interior angles of any quadrilateral is 360º.

Types of Quadrilaterals

Quadrilaterals occur in several special forms. Common types and their defining properties are given below.

Parallelogram

A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In parallelogram ABCD, AB ∥ CD and AD ∥ BC. Opposite sides of a parallelogram are equal and opposite angles are equal.

Rectangle

A parallelogram in which each interior angle is 90º is called a rectangle. All rectangles have diagonals that bisect each other and are equal in length.

Square

A rectangle with all four sides equal is called a square. A square has all properties of a rectangle and a rhombus: diagonals are equal, bisect each other at right angles, and bisect the interior angles.

Rhombus

A parallelogram with all four sides equal is called a rhombus. Diagonals of a rhombus bisect each other at right angles and each diagonal bisects a pair of opposite angles.

Trapezium

A quadrilateral in which exactly one pair of opposite sides is parallel is called a trapezium (also called trapezoid in some texts). If the non-parallel sides of a trapezium are equal, it is called an isosceles trapezium.

Kite

A quadrilateral with two distinct pairs of adjacent equal sides is called a kite. In kite PQRS, PQ = QR and PS = RS. Diagonals of a kite are perpendicular and one diagonal bisects the other.

Remarks:

• A square, rectangle and rhombus are all parallelograms.
• A square is both a rectangle and a rhombus, but a rectangle (or a rhombus) need not be a square.
• A parallelogram is not always a trapezium; a trapezium need not be a parallelogram.
• A kite is not a parallelogram in general.

Properties of Parallelograms

• A diagonal of a parallelogram divides it into two congruent triangles.
• In a parallelogram, opposite sides are equal.
• In a parallelogram, opposite angles are equal.
• The diagonals of a parallelogram bisect each other.

The Mid-Point Theorem

Statement: The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Proof: Consider triangle ABC with D the mid-point of AB and E the mid-point of AC. Join DE.
AD = DB because D is the mid-point of AB.
AE = EC because E is the mid-point of AC.
AD/AB = AE/AC = 1/2.
Angle A is common to triangles ADE and ABC.

Therefore, triangles ADE and ABC are similar (corresponding sides are in the same ratio and included angle are equal).
From similarity, DE/BC = AD/AB = 1/2.
Hence DE = (1/2)·BC and DE ∥ BC.

Thus the segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.

Converse of the Mid-Point Theorem

Statement: A line drawn through the mid-point of one side of a triangle and parallel to a second side meets the third side at its mid-point.

Proof:
Let ABC be a triangle and let D be the mid-point of AB. Through D draw a line DE parallel to BC meeting AC at E.
Since DE ∥ BC, triangles ADE and ABC are similar (angle A is common and corresponding angles are equal).
From similarity, AD/AB = AE/AC.
Because AD = (1/2)·AB, we get AE = (1/2)·AC.
Therefore E is the mid-point of AC.
Hence the line through the mid-point of one side and parallel to another side meets the third side at its mid-point.