A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 +...

QUADRILATERALS

CH - 8

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A Quadrilateral is a figure which is formed by joining four points in an order is called a quadrilateral.

Each figure is formed by Joining 4 points.


Sides – 4

A

B C

D

Angles sum – 360

Diagonals - 2


Angle Sum Property of a Quadrilateral

Let ABCD be a quadrilateral. Join AC.

Clearly, 1 + 2 = A ...... (i) ∠ ∠ ∠

And, 3 + 4 = C ...... (ii) ∠ ∠ ∠

We know that the sum of the angles of a triangle is 180°.

Angle Sum Property of a Quadrilateral

Therefore, from ∆ABC, we have

∠2 + 4 + B = 180° (Angle sum property of triangle)∠ ∠

From ∆ACD, we have

∠1 + 3 + D = 180° (Angle sum property of triangle) ∠ ∠

Adding the angles on either side, we get;

∠2 + 4 + B + 1 + 3 + D = 360° ∠ ∠ ∠ ∠ ∠

⇒ ( 1 + 2) + B + ( 3 + 4) + D = 360° ∠ ∠ ∠ ∠ ∠ ∠

⇒ ∠A + B + C + D = 360° [using (i) and (ii)]. ∠ ∠ ∠

Hence, the sum of all the four angles of a quadrilateral is 360°.

It states that sum of angles of a quadrilaterals is 360⁰PROOF :-


Major Types of Quadrilaterals


PARALLELOGRAM

• Opposite sides are parallel by definition.

• Opposite sides are congruent.

• Opposite angles are congruent.

• Consecutive angles are supplementary.

• The diagonals bisect each other.


TRAPEZIUM

• One pair of opposite side are parallel.

• Opposite non-parallel sides are equal in length.


SQUAREA D

B C

•The diagonals of a square bisect each other and meet at 90°•The diagonals of a square bisect its angles.•The diagonals of a square are perpendicular.

•Opposite sides of a square are both parallel and equal in length.•All four angles of a square are equal.•All four sides of a square are equal.


RHOMBUS

•All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary).•All sides are congruent by definition.•The diagonals bisect the angles.


Rectangle

•All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other).•All angles are right angles by definition.•The diagonals are congruent


KITE

• The diagonals of a kite meet at a right angle.

• Kites have exactly one pair of opposite angles that are congruent.

• The Opposite sides are parallel.


Note-A square, rectangle and rhombus are all parallelograms.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.


----------Theorems----------• A diagonal of a parallelogram divides it

into two congruent triangles.• In a parallelogram, opposite sides are

equal.• If each pair of opposite sides of a

quadrilateral is equal, then it is a parallelogram.

• In a parallelogram, opposite angles are equal.

• If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

• The diagonals of a parallelogram bisect each other.

• If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

• A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.


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A BC D Let ABCD be a quadrilateral. Join AC. Clearly, ∠ 1 + ∠ 2 = ∠ A...... (i) And, ∠ 3 +...

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