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Quadrilaterals | Speed Notes
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Quadrilateral
Any closed polygon with four sides, four angles and four vertices are called Quadrilateral. It could be regular or irregular. (Sroll down to continute …)
![Quadrilateral Quadrilateral](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-43.png?resize=293%2C182&ssl=1)
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Revision Notes – CBSE 09 Math – Quadrilaterals
Angle Sum Property of a Quadrilateral
The sum of the four angles of a quadrilateral is 360°
![Angle Sum Property of a Quadrilateral Angle Sum Property of a Quadrilateral](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-39.png?resize=220%2C156&ssl=1)
If we draw a diagonal in the quadrilateral, it divides it into two triangles.
And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.
The sum of angles of ∆ADC = 180°.
The sum of angles of ∆ABC = 180°.
By adding both we get ∠A + ∠B + ∠C + ∠D = 360°
Hence, the sum of the four angles of a quadrilateral is 360°.
Example
Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.
![quadrilateral ABCD quadrilateral ABCD](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-40.png?resize=338%2C147&ssl=1)
Solution:
Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.
So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).
∠B = 52°
∠A = 180°- 52° = 128°
∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).
∠C = 60°
128° + 52° + 60° + ∠D = 360°
∠D = 120°
∴ ∠A = 128° and ∠D = 120 °.
Types of Quadrilaterals
S No. | Quadrilateral | Property | Image |
1. | Trapezium | One pair of opposite sides is parallel. | |
2. | Parallelogram | Both pairs of opposite sides are parallel. | |
3. | Rectangle | a. Both the pair of opposite sides is parallel.b. Opposite sides are equal.c. All the four angles are 90°. | |
4. | Square | a. All four sides are equal.b. Opposite sides are parallel.c. All the four angles are 90°. | |
5. | Rhombus | a. All four sides are equal.b. Opposite sides are parallel.c. Opposite angles are equal.d. Diagonals intersect each other at the centre and at 90°. | |
6. | Kite | Two pairs of adjacent sides are equal. |
Remark: A square, Rectangle and Rhombus are also a parallelogram.
Properties of a Parallelogram
Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.
![Parallelogram Parallelogram](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-41.png?resize=424%2C179&ssl=1)
∆ABD ≅ ∆CDB
Theorem 2: In a parallelogram, opposite sides will always be equal.
![In a parallelogram, opposite sides will always be equal In a parallelogram, opposite sides will always be equal](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-35.png?resize=304%2C176&ssl=1)
Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.
![A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal. A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-33.png?resize=228%2C136&ssl=1)
Here, AD = BC and AB = DC
Then ABCD is a parallelogram.
Theorem 4: In a parallelogram, opposite angles are equal.
![In a parallelogram, opposite angles are equal. In a parallelogram, opposite angles are equal.](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-42.png?resize=244%2C155&ssl=1)
In ABCD, ∠A = ∠C and ∠B = ∠D
Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.
Theorem 6: The diagonals of a parallelogram bisect each other.
![The diagonals of a parallelogram bisect each other. The diagonals of a parallelogram bisect each other.](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-37.png?resize=211%2C212&ssl=1)
Here, AC and BD are the diagonals of the parallelogram ABCD.
So the bisect each other at the centre.
DE = EB and AE = EC
Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.
This is the reverse of the theorem 6.
The Mid-point Theorem
1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.
![Triangle Triangle](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-36.png?resize=313%2C185&ssl=1)
If AB = BC and CD = DE then BD ∥ AE.
2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line.
![Triangle Triangle](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-34.png?resize=203%2C182&ssl=1)
If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.
Example
Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.
![Trapezium Trapezium](https://i0.wp.com/therightmentor.com/wp-content/uploads/2023/11/image-38.png?resize=313%2C163&ssl=1)
Solution:
Let BE cut DC at a point G.
Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.
By midpoint theorem, G is the midpoint of EB.
Again in ∆BEF, G is the midpoint of BE and GC∥ EF.
So, by midpoint theorem C is the midpoint of BF.
Hence proved.
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