A quadrilateral has four sides, four angle and four vertices. In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices and ∠A, ∠B, ∠C and ∠D are the four angles formed at the vertices.
If we join opposite vertices then we get AC and BD which are the two diagonals of the quadrilateral.
Angle Sum Property of Quadrilateral
The sum of the angles of a quadrilateral is 360º.
This can be verified by drawing a diagonal and dividing the quadrilateral into two triangles.
We know that
∠ DAC + ∠ ACD + ∠ D = 180° (1)
Similarly, in Δ ABC,
∠ CAB + ∠ ACB + ∠ B = 180° (2)
Adding (1) and (2), we get
∠ DAC + ∠ ACD + ∠ D + ∠ CAB + ∠ ACB + ∠ B = 180° + 180° = 360°
Also, ∠ DAC + ∠ CAB = ∠ A and ∠ ACD + ∠ ACB = ∠ C
So, ∠ A + ∠ D + ∠ B + ∠ C = 360°.
i.e., the sum of the angles of a quadrilateral is 360°.
Types of Quadrilaterals
- One pair of opposite sides of quadrilateral ABCD in namely, AB and CD are parallel. It is called a trapezium.
- Both pairs of opposite sides of quadrilaterals given in Fig (ii), (iii) , (iv) and (v) are parallel. Such quadrilaterals are called parallelograms. So, quadrilateral PQRS of Fig (ii) is a parallelogram.
- In parallelogram MNRS of Fig (iii), note that one of its angles namely ∠ M is a right angle. It is called a rectangle.
- The parallelogram DEFG of Fig (iv) has all sides equal and it is called a rhombus.
- The parallelogram ABCD of Fig (v) has ∠ A = 90° and all sides equal; it is called a square.
- In quadrilateral ABCD of Fig (vi), AD = CD and AB = CB i.e., two pairs of adjacent sides are equal. It is not a parallelogram. It is called a kite.
Properties of Parallelogram
Axiom 1: A diagonal of a parallelogram divides it into two congruent triangles.
Proof: Let ABCD be a parallelogram and AC be a diagonal .Observe that the diagonal AC divides parallelogram ABCD into two triangles, namely, Δ ABC and Δ CDA. We need to prove that these triangles are congruent.
In Δ ABC and Δ CDA, note that BC || AD and AC is a transversal.
So, ∠ BCA = ∠ DAC (Pair of alternate angles)
Also, AB || DC and AC is a transversal.
So, ∠ BAC = ∠ DCA (Pair of alternate angles)
and AC = CA (Common)
So, Δ ABC ≅ Δ CDA (ASA rule)
Axiom 2: In a parallelogram, opposite sides are equal.
Axiom 3: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Axiom 4: In a parallelogram, opposite angles are equal.
Axiom 5: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Axiom 6: The diagonals of a parallelogram bisect each other.
Axiom 7: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.