Quadrilaterals

A quadrilateral is a polygon with four sides and four angles. The sum of the interior angles of any quadrilateral is always \(360^\circ\). This chapter explores various types of quadrilaterals, their properties, and applications.

Formula Derivation: Sum of Interior Angles

Consider a quadrilateral \(ABCD\). By drawing a diagonal \(AC\), the quadrilateral is divided into two triangles \(\triangle ABC\) and \(\triangle ADC\). Since the sum of interior angles of a triangle is \(180^\circ\), the sum of interior angles of the quadrilateral is:

\[ \angle ABC + \angle BCD + \angle CDA + \angle DAB = 180^\circ + 180^\circ = 360^\circ \]

Types of Quadrilaterals and Their Properties

• Rectangle: Four right angles (\(90^\circ\)), opposite sides equal and parallel, diagonals equal and bisect each other.
• Square: All sides equal, four right angles, diagonals equal, perpendicular, and bisect the angles.
• Parallelogram: Opposite sides equal and parallel, opposite angles equal, adjacent angles sum to \(180^\circ\), diagonals bisect each other.
• Rhombus: All sides equal, opposite sides parallel, diagonals bisect each other at \(90^\circ\) and bisect the angles.
• Trapezium: At least one pair of parallel sides; if isosceles, non-parallel sides and base angles are equal.
• Kite: Two pairs of adjacent sides equal, one diagonal bisects the other at \(90^\circ\), bisects a pair of opposite angles.

Worked Illustration

Verify that the sum of interior angles of a quadrilateral is \(360^\circ\).

Step 1: Draw diagonal \(AC\) dividing quadrilateral \(ABCD\) into two triangles.

Step 2: Sum of angles in \(\triangle ABC = 180^\circ\).

Step 3: Sum of angles in \(\triangle ADC = 180^\circ\).

Step 4: Total sum = \(180^\circ + 180^\circ = 360^\circ\).

Solved Example

Example: In parallelogram \(ABCD\), if \(\angle A = 70^\circ\), find the other angles.

Solution:

Step 1: Opposite angles are equal, so \(\angle C = 70^\circ\).

Step 2: Adjacent angles sum to \(180^\circ\), so \(\angle B = 180^\circ - 70^\circ = 110^\circ\).

Step 3: Opposite angles equal, so \(\angle D = 110^\circ\).

Practice Set

• Level 1 – Easy
• Find the sum of interior angles of a quadrilateral.
• Identify the number of right angles in a rectangle.
• State the properties of a square.
• Level 2 – Moderate
• In a rhombus, if one angle is \(60^\circ\), find the other angles.
• Prove that the diagonals of a rectangle are equal.
• Calculate the measure of adjacent angles in a parallelogram if one angle is \(85^\circ\).
• Level 3 – Challenging
• Prove that the diagonals of a rhombus bisect each other at right angles.
• In an isosceles trapezium, prove that the base angles are equal.
• Given a kite with sides \(AB = AD = 5\) cm and \(BC = CD = 7\) cm, find the length of the diagonal that bisects the other.

Answer Key

• Level 1
• Sum of interior angles = \(360^\circ\).
• Rectangle has 4 right angles.
• Square has all sides equal and 4 right angles.
• Level 2
• Opposite angles of rhombus are equal, so other angles are \(120^\circ\).
• Diagonals of rectangle are equal because it is a parallelogram with right angles; proof involves congruent triangles.
• Adjacent angles = \(180^\circ - 85^\circ = 95^\circ\).
• Level 3
• Proof involves showing triangles formed by diagonals are congruent and angles are right angles.
• Base angles equal due to isosceles trapezium properties and congruent triangles.
• Use Pythagoras theorem on kite diagonals to find length.

Quick Reference

Quadrilateral	Properties
Rectangle	4 right angles, opposite sides equal, diagonals equal and bisect each other
Square	All sides equal, 4 right angles, diagonals equal, perpendicular, bisect angles
Parallelogram	Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other
Rhombus	All sides equal, diagonals bisect at \(90^\circ\), bisect angles
Trapezium	At least one pair of parallel sides; isosceles trapezium has equal non-parallel sides and base angles
Kite	Two pairs of adjacent equal sides, one diagonal bisects the other at \(90^\circ\)

Glossary

• Quadrilateral: A polygon with four sides and four angles.
• Diagonal: A line segment joining two non-adjacent vertices.
• Parallelogram: A quadrilateral with opposite sides parallel.
• Rhombus: A parallelogram with all sides equal.
• Rectangle: A parallelogram with four right angles.
• Square: A rectangle with all sides equal.
• Trapezium: A quadrilateral with at least one pair of parallel sides.
• Kite: A quadrilateral with two pairs of adjacent equal sides.