Mathematics Notes for Class 11 Quadrilaterals

Quadrilaterals

Welcome to the chapter on Quadrilaterals for Class 11. In this chapter, you will learn about different types of quadrilaterals, their properties, important theorems, and how to solve problems involving quadrilaterals. By the end of this chapter, you will be able to identify, classify, and prove properties related to quadrilaterals.

Key Concepts

Quadrilateral: A polygon with four sides, four vertices, and four angles.
Types of Quadrilaterals: Parallelogram, rectangle, square, rhombus, trapezium, kite.
Diagonals: Line segments joining opposite vertices.
Sum of Angles: The sum of the interior angles of a quadrilateral is 360°.

Types of Quadrilaterals

Parallelogram: Opposite sides are parallel and equal; opposite angles are equal.
Rectangle: All angles are 90°; opposite sides are equal and parallel.
Square: All sides are equal; all angles are 90°.
Rhombus: All sides are equal; opposite angles are equal; diagonals bisect at right angles.
Trapezium: Only one pair of opposite sides is parallel.
Kite: Two pairs of adjacent sides are equal; one pair of opposite angles is equal.

Properties of Quadrilaterals

The sum of the four interior angles is always 360°.
The diagonals of a parallelogram bisect each other.
The diagonals of a rectangle are equal and bisect each other.
The diagonals of a rhombus bisect each other at right angles.
The diagonals of a square are equal and bisect each other at right angles.

Important Theorems

Theorem 1: The diagonals of a parallelogram bisect each other.
Theorem 2: In a rectangle, the diagonals are equal and bisect each other.
Theorem 3: In a rhombus, the diagonals bisect each other at right angles.
Theorem 4: The sum of the angles of a quadrilateral is 360°.

Applications

Finding unknown angles and sides in quadrilaterals.
Proving properties using congruence and parallel lines.
Solving real-life problems involving land plots, tiles, and design.

Practice Questions

Prove that the diagonals of a rectangle are equal and bisect each other.
If the angles of a quadrilateral are in the ratio 2:3:4:5, find all the angles.
Show that the diagonals of a rhombus bisect each other at right angles.
In a parallelogram ABCD, if ∠A = 70°, find the other angles.
Explain why a square is both a rectangle and a rhombus.

Challenge Yourself

Draw a quadrilateral that is not a parallelogram. Mark its sides and angles.
Given a quadrilateral with vertices A(0,0), B(4,0), C(3,3), D(1,3), prove it is a trapezium.
If the diagonals of a quadrilateral bisect each other, is it always a parallelogram? Justify your answer.

Did You Know?

A square is a special type of rectangle and rhombus.
The word "quadrilateral" comes from Latin, meaning "four sides".

Glossary

Quadrilateral: A polygon with four sides.
Diagonal: A line joining two non-adjacent vertices.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Trapezium: A quadrilateral with only one pair of parallel sides.

Answers to Practice Questions

By using congruent triangles and properties of parallel lines, we can show the diagonals of a rectangle are equal and bisect each other.
Let the angles be 2x, 3x, 4x, 5x. 2x+3x+4x+5x=360° ⇒ 14x=360° ⇒ x=25°. So, angles are 50°, 75°, 100°, 135°.
By drawing the diagonals and using triangle congruence, we can prove they bisect at right angles.
Opposite angles of a parallelogram are equal. ∠A=70°, ∠C=70°, ∠B=110°, ∠D=110°.
A square has all sides equal (like a rhombus) and all angles 90° (like a rectangle).

Master the properties and theorems of quadrilaterals to solve advanced geometry problems!

Mathematics Class 11 - Quadrilaterals Notes