In the NCERT Maths Class 9 for Quadrilaterals, concepts are properly taught from the basic explanation of quadrilaterals to a variety of axioms and formulae that prove their connection to other figures. Parallelogram properties, quadrilateral forms and angle sum properties are among some of the central topics of this chapter. Quadrilaterals are one of the most important topics in the Class 9 Mathematics syllabus. Let us understand it better with the help of this blog!
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What is a Quadrilateral?
The figure formed by combining the order of four points is called a quadrilateral. A quadrilateral has the following characteristics:
1. Four sides
2. Four angles
3. Four vertices
For example, 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. Some of the real time examples of quadrilaterals are – the floor, walls, ceiling, windows of your classroom, the blackboard, each face of the duster, each page of your book, the top of your study table, etc.
Angle Sum Property of Quadrilaterals Class 9
The sum of the angles of a quadrilateral is 360°
Take the same quadrilateral ABCD that is given below. You 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°
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Here are some other properties that constitute a quadrilateral:
- A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
- The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
- A line through the mid- point of a side of a triangle parallel to another side bisects the third side.
Types of Quadrilaterals
- A square is a rectangle and also a rhombus
- A parallelogram is a trapezium
- A kite is not a parallelogram
- A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram)
- A rectangle or a rhombus is not a square
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What is a parallelogram?
In simple terms, a parallelogram is a simple quadrilateral containing two pairs of parallel sides. The opposite or opposite sides of the parallelogram are also of equal length as well as the opposite angles of the parallelogram are of equal measure.
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Properties of a Parallelogram Class 9
First Property: A diagonal of a parallelogram divides it into two congruent triangles
Proof : Let ABCD be a parallelogram and AC be a diagonal (see Fig. 8.8).
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) or, diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA.
Now, measure the opposite sides of parallelogram ABCD
You will get AB = DC and AD = BC
This is another property of a parallelogram stated below:
Second Property: In a parallelogram, opposite sides are equal.
Proof: According to the first property, you get AB = DC and AD = BC which proves the second property.
Third Property: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Proof: Let sides AB and CD of the quadrilateral ABCD be equal and also AD = BC.
Draw diagonal AC.
Clearly, Δ ABC ≅ Δ CDA
So, ∠ BAC = ∠ DCA and ∠ BCA = ∠ DAC
Using the above theorems, some other theorems can also be derived. They are as follows:
Fourth Property: In a parallelogram, opposite angles are equal
Fifth Property: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram
Sixth Property: The diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Hence, these are some of the properties present in a parallelogram. Here are some example problems for you to understand:
Example 1: Show that the diagonals of a rhombus are perpendicular to each other.
Consider the rhombus ABCD
You know that AB = BC = CD = DA
Now, in Δ AOD and Δ COD, OA = OC (Diagonals of a parallelogram bisect each other)
OD = OD (Common) AD = CD
Therefore, Δ AOD ≅ Δ COD (SSS congruence rule)
This gives, ∠ AOD = ∠ COD (CPCT)
But, ∠ AOD + ∠ COD = 180° (Linear pair)
So, 2∠ AOD = 180° or, ∠ AOD = 90°
So, the diagonals of a rhombus are perpendicular to each other.
These were the class 9 quadrilaterals study notes and properties. Do you really enjoy solving Maths problems? Is Maths your favourite subject? Want to know what are the popular courses and universities for any particular subject? Get in touch with Leverage Edu experts to find out!