CBSE Class 9 Maths Revision Notes Chapter 8

Class 9 Mathematics Revision Notes for Quadrilaterals of Chapter 8

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Class 9 Mathematics Revision Notes for Quadrilaterals of Chapter 8

Access Class 9 Mathematics Chapter 8 – Quadrilaterals Notes

Parallelograms:

• An equal and opposite parallel pair of sides make up a parallelogram, a quadrilateral.
• Rectangles, rhombuses, and squares make up parallelograms.
• A trapezium is a quadrilateral with a single pair of parallel opposite sides. It does not consequently create a parallelogram.
• All pairs have equal and parallel opposite sides.

Opposite Sides of a Quadrilateral:

• Opposite sides are two sides of a quadrilateral that share no common point.

• AB and DC are one pair of opposite sides in the diagram.

• The other pair of opposite sides is DC and BC.

Consecutive Sides of a Quadrilateral:

• Consecutive sides are two sides of a quadrilateral that share an endpoint.

• As shown in the diagram,

• AB and BC are two consecutive sides.

• The other three sets of successive sides are BC, CD; CD, DA; and DA, AB.

Opposite Angles of a Quadrilateral:

• The opposite angles of a quadrilateral are two angles that intersect but do not comprise a side.

• As shown in the diagram,

• One set of opposite angles is ∠A and ∠C, while another pair of opposite angles is ∠B and ∠D.

Consecutive Angles of a Quadrilateral:

• The intersection of two quadrilateral angles that comprise a side is known as a consecutive angle.

• As shown in the diagram,

• ∠A and ∠B are one pair of successive angles, while the other three pairs are ∠B, ∠C; ∠C, ∠D; and ∠D, ∠A.

Sufficient Conditions for a Quadrilateral to be a Parallelogram:

• The following is the defining property of a parallelogram:
• “The opposite sides of a quadrilateral are equal if it is a parallelogram.”

Special Parallelograms:

Parallelograms are made up of rectangles, rhombuses, and squares.

Here are some possible definitions for each of these:

• A is an equilateral and equiangular parallelogram. As a result, a square can be a rectangle as well as a rhombus.
• The graphic below depicts the relationships between the special parallelograms:

• As every rectangle and rhombus would be a parallelogram, they are represented as subsets of a parallelogram, while a square is represented by the overlapping shaded area because it is both a rectangle and a rhombus.

Rectangle:

A rectangle is a parallelogram with a right angle at one of its corners.

In the above figure,

Let, ∠A=90∘

Since,

AD∥BC,

∠A+ ∠B=180∘

(Sum of interior angles along the same transverse AB side)

Therefore,

∠B=90∘

Here,

AB∥CD and ∠A=90∘ (Given)

Therefore,

∠A+∠D=180∘

   ∴∠D=90∘

   ∴∠C=90∘

Corollary: Each of a rectangle’s four angles is a right angle.

Rhombus:

A parallelogram with two successive equal sides is known as a rhombus.

ABCD  is a rhombus in which AB=BC.

Since a rhombus is a parallelogram, 

AB=DC and BC=AD

 Thus, AB=BC=CD=AD

Corollary: A rhombus’s four sides are all equal (congruent).

Square:

A square is a rectangle that has two sides of the same length on consecutive sides.

Since a square is a rectangle, 

AB=DC and BC=CD are true since every angle of a rectangle is a right angle.

So, AB=BC=CD=AD.

A square has four equal-length sides and four right angles, all of which are the same length.

Class 9 Mathematics Quadrilaterals Notes Free 

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Class 9 Mathematics Quadrilaterals Notes

The chapter begins with a review of triangles which are made up of three angles. Similarly, a quadrilateral is a shape made up of four angles and four sides of varying lengths. These figures have specific qualities, which will be explored further in the chapter. Certain objects, such as a chalkboard, book, or table, can be used as examples for quadrilaterals to help students understand these figures.

• Polygons are planar figures with sides defined by straight-line segments that we are all familiar with.

• The term polygon is derived from Greek.

• It refers to a figure with many angles, implying many sides.

• Quadrilaterals are four-sided geometric shapes formed by the union of four line segments.

• A quadrilateral is a four-sided polygon.

Angle Sum Property of The Quadrilateral

In this section, one of the basic features of the quadrilateral ‘additive property’ is explained. According to this property, the total of all quadrilateral angles equals 360°. It denotes a complete circle angle.

Types of Quadrilaterals

Students are familiar with a quadrilateral and its angle sum property. In this section, they will learn more about quadrilaterals and their different varieties. According to Chapter 8 of Class 9 Mathematics Notes, there are five major varieties of quadrilaterals. They are: 

• The trapezium is a quadrilateral figure that has a single pair of parallel sites.
• A parallelogram is a form of a quadrilateral in which both pairs are parallel to one another.
• A rectangle is the third type of quadrilateral. It has four right angles and two equal corresponding sides.
• A rhombus is a quadrilateral with all four sides equal.
• A square is a quadrilateral that includes the characteristics of both a rectangle and a rhombus. It has four equal sides and 90° angles.

Properties of Parallelogram

This section explains the different properties of parallelograms with theorems and proofs. A parallelogram’s attributes are as follows:

• A parallelogram’s diagonal can be used to divide into two congruent triangles.
• The opposite sides of a parallelogram are equal.
• It is similar to the previous property. If the opposite sides of a quadrilateral are equal, the quadrilateral is said to be a parallelogram.
• The parallelogram has another property known as opposite angles equal.
• If two diagonals in a parallelogram are drawn, they will bisect each other.
• The quadrilateral is known as a parallelogram if the above-mentioned criterion is satisfied and the diagonals are likewise bisected.
• When the diagonals are drawn in a rhombus, they are perpendicular to each other.

Another Condition For A Quadrilateral To Be A Parallelogram

Another criterion for a quadrilateral to be a parallelogram is that if a figure of quadrilaterals with opposite sides is equal and parallel to each other, it is a parallelogram. So far, it was assumed that any of the conditions can be met; however, there is now an extra condition that meets both criteria.

Midpoint Theorem

In this final section, students are introduced to a new theorem called the midpoint theorem. It explains that if a line segment connects two sides of a triangle; it is parallel to the third side of the triangle. A diagram makes it simple to demonstrate this. Class 9 Mathematics Chapter 8 Notes additionally provides an explanation of the converse of the theorem. This theorem is also known as the exact midpoint theorem. This theorem states that the line extended to the middle of one triangle’s side is parallel to the other side and cuts the third side in half.