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Quadrilaterals – Class 9 Mathematics notes, important questions and MCQs

NCERT Class 9 Mathematics, chapter 12. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A quadrilateral has four sides and its angles add up to 360°. Parallelograms and their special types (rectangle, rhombus, square) have useful properties of sides, angles and diagonals.

Angle sum and types

Sum of interior angles = 360°.
Trapezium: one pair of parallel sides; kite: two pairs of equal adjacent sides.
Parallelogram: both pairs of opposite sides parallel.

Properties of a parallelogram

Opposite sides are equal and opposite angles are equal.
Diagonals bisect each other; a diagonal divides it into two congruent triangles.
Adjacent angles are supplementary (add to 180°).

Special parallelograms

Rectangle: all angles 90°, diagonals equal.
Rhombus: all sides equal, diagonals bisect each other at 90°.
Square: all sides equal, all angles 90°, diagonals equal and perpendicular.
Mid-point theorem: the segment joining mid-points of two sides of a triangle is parallel to the third side and half of it.

Key terms

Parallelogram: quadrilateral with both pairs of opposite sides parallel
Rhombus: parallelogram with all sides equal
Mid-point theorem: segment joining mid-points of two sides is parallel to and half the third side

Formulas

Angle sum of a quadrilateral = 360°
Adjacent angles of a parallelogram add to 180°

Important questions with answers

The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find them.2 marks

Sum = 360°, so 30x = 360°, x = 12°. Angles: 36°, 60°, 108°, 156°.

Show that the diagonals of a rhombus bisect each other at right angles.3 marks

A rhombus is a parallelogram, so its diagonals bisect each other at O. In △AOB and △COB: AB = CB (sides of rhombus), AO = CO, OB common, so they are congruent (SSS). Hence ∠AOB = ∠COB, and as they form a linear pair, each is 90°.

State and prove the mid-point theorem.5 marks

Statement: the line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. Proof: In △ABC, let E and F be mid-points of AB and AC. Extend EF to D so that EF = FD and join CD. In △AEF and △CDF: AF = CF, EF = DF, ∠AFE = ∠CFD (vertically opposite), so they are congruent (SAS). Hence AE = CD and ∠EAF = ∠DCF, which makes CD ∥ AB. So BE = AE = CD and BE ∥ CD, making EBCD a parallelogram. Therefore ED ∥ BC and ED = BC; since EF = ½ED, EF ∥ BC and EF = ½BC.

Quick revision

Diagonals of a parallelogram bisect each other.
Rhombus diagonals are perpendicular.
Mid-point segment = half the third side.

Exam tips

To prove a parallelogram, show one pair of opposite sides is equal and parallel.
Every square is a rhombus and a rectangle, but not the other way round.

Practice MCQs

Sum of the angles of a quadrilateral is
1. 180°
2. 270°
3. 360°
4. 540°
Show answer
Answer: (c) 360°. Two triangles: 2 × 180°.
Diagonals of a rectangle are
1. equal
2. perpendicular
3. unequal
4. parallel
Show answer
Answer: (a) equal. They are equal.
A quadrilateral with all sides equal and all angles 90° is a
1. square
2. rectangle
3. rhombus
4. kite
Show answer
Answer: (a) square. Square.
In a parallelogram, one angle is 70°. Its adjacent angle is
1. 70°
2. 90°
3. 110°
4. 20°
Show answer
Answer: (c) 110°. Adjacent angles add to 180°.
Three angles of a quadrilateral are 75°, 90°, 75°. The fourth is
1. 120°
2. 90°
3. 100°
4. 110°
Show answer
Answer: (a) 120°. 360 − 240.
Diagonals of a rhombus meet at
1. 45°
2. 60°
3. 90°
4. 120°
Show answer
Answer: (c) 90°. They are perpendicular.
In △ABC, D and E are mid-points of AB and AC. If BC = 14 cm, then DE =
1. 7 cm
2. 14 cm
3. 28 cm
4. 3.5 cm
Show answer
Answer: (a) 7 cm. Mid-point theorem.
The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. The smallest angle is
1. 45°
2. 30°
3. 36°
4. 60°
Show answer
Answer: (c) 36°. 30 parts = 360°, 1 part = 12°, smallest = 36°.

All chapters of Class 9 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.