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Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9 PDF Download

  1. A plane figure bounded by four sides is called a quadrilateral.
  2. The sum of all the interior angles of a quadrilateral is 360º.
  3. A quadrilateral is a parallelogram, if
    (i) opposite sides are equal,
    (ii) opposite angles are equal,
    (iii) diagonals bisect each other,
    (iv) a pair of opposite sides is equal and parallel.
  4. A diagonal of a parallelogram divides it into two congruent triangles.
  5. Diagonals of a rhombus bisect each other at right angles, and vice versa.
  6. Diagonals of a square bisect each other at right angles and are equal and vice versa.
  7. Diagonals of a rectangle bisect each other and vice versa.
  8. A line segment joining the mid-points of any two sides of a triangle is parallel to the third side.
  9. A line segment joining the mid-points of any two sides of a triangle is half of the third side.
  10. A line through the mid-point of a side of a triangle, parallel to another side bisects the third side.
  11. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral is a parallelogram.

Quadrilateral

A plane figure bounded by four sides is called a quadrilateral. In the following figure, four-line segments AB, BC, CD and DA bound a quadrilateral ABCD.
In the given figure, we have.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9

(i) The points A, B, C and D are the vertices of quadrilateral ABCD.
(ii) The line segments AB, BC, CD and DA are the sides of quadrilateral ABCD.
(iii) The line segments AC and BD are called the diagonals of quadrilateral ABCD.

Note: 
(i) Two sides having a common end point are called adjacent sides.
(ii) Two sides having no common end point are called opposite sides.
(iii) Two angles of a quadrilateral having a common arm are called consecutive angles.
(iv) Two angles of a quadrilateral having no common arm are called its opposite angles.

Angle Sum Property of a Quadrilateral

The sum of all the four angles of a quadrilateral is 360º

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9

Let us prove it:
In a quadrilateral ABCD, join BD. Since, the sum of the angles of a triangle is 180º.
∴ In ΔABD, we have
∠1 + ∠A + ∠2 = 180º     …(1)
In ΔBCD, we have
∠3 + ∠C + ∠4 = 180º    …(2)
Adding (1) and (2), we get
[∠1 + ∠A + ∠2] + [∠3 + ∠C + ∠4] = 180º + 180º
(∠1 + ∠3) + ∠A + ∠C + (∠2 + ∠4) = 180° + 180°
⇒ ∠B + ∠A + ∠C + ∠D = 360º  
[∵ ∠1 + ∠3 = ∠B and ∠2 + ∠4 = ∠D]
∴ ∠A + ∠B + ∠C + ∠D = 360º
Thus, the sum of four angles of a quadrilateral is 360º.

Types of Quadrilaterals

The various types of quadrilaterals are:
(i) Parallelogram:

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9

A quadrilateral in which opposite sides are parallel is called a parallelogram. In the figure, ABCD is a parallelogram. Here, AB || CD and AD || BC.
Also, opposite sides of a parallelogram are equal.
(ii) Rectangle: A parallelogram, each of whose angle is 90º, is called a rectangle.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9
In the figure, PQRS is a rectangle. We write it as rectangle PQRS.
(iii) Square: A rectangle having all sides equal is called a square. In the figure, LMNO is a square.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9
(iv) Rhombus: A parallelogram having all sides equal is called a rhombus. In the figure, PQRS is a rhombus.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9
(v) Trapezium: A quadrilateral in which two opposite sides are parallel and two opposite sides are nonparallel, is called a trapezium. In the figure, BCDE is a trapezium.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9

Note: 
If the two non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium.

(vi) Kite: A quadrilateral in which two pairs of adjacent sides are equal is known as kite. PQRS is a kite such that PQ = QR and PS = RS.

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9

Note: 
(i) A square, rectangle and rhombus are all parallelograms.
(ii) A square is a rectangle and also a rhombus, but a rectangle or a rhombus is not a square.
(iii) A parallelogram is a trapezium, but a trapezium is not a parallelogram.
(iv) A kite is not a parallelogram.

Properties of Parallelograms

1. A diagonal of a parallelogram, divides it into two congruent triangles.
2. In a parallelogram, opposite sides are equal.
3. In a parallelogram, opposite angles are equal.
4. The diagonals of a parallelogram bisect each other.

The Mid-Point Theorem

Statement: “The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.”
Proof: We have ΔABC in which D and E are the mid-points of AB and AC respectively. Let us join DE and produce it. Draw CF || BA to meet DE (produced) in F.
We have to prove that DE = (1/2)BC and DE || BC.
Now, in ΔAED and ΔCEF, we have
AE = CE   [Given]
∠AED = ∠CEF   [Vertically opposite angles]
∠DAE = ∠FCE    [∵ AB || CF and AC is a transversal]
∴ ΔAED ≌ ΔCEF  [ASA criteria]

Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9
⇒ Their corresponding parts are equal.
⇒ AD = CF and DE = EF
But AD = BD    [∵ D is mid point of AB]
⇒ BD = CF Also, BD || CF   [By construction]
⇒ BCFD is a parallelogram.
∴ DF = BC and DF || BC
⇒ (1/2)(DF) = (1/2)
(BC) and DE || BC    [∵ DE = EF]
⇒ DE = (1/2)
BC and DE || BC Hence, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Converse of Mid-Point Theorem

The line drawn through the mid-point of one side of a triangle parallel to another side intersects the third side at its mid-point.

The document Facts that Matter: Quadrilaterals | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Facts that Matter: Quadrilaterals - Mathematics (Maths) Class 9

1. What is the Mid-Point Theorem?
Ans. The Mid-Point Theorem states that in a quadrilateral, the line segment joining the midpoints of two sides is parallel to the line segment joining the midpoints of the other two sides.
2. How is the Mid-Point Theorem useful in quadrilaterals?
Ans. The Mid-Point Theorem is useful in quadrilaterals as it helps in proving the parallelism of certain line segments within the quadrilateral. It allows us to establish relationships between the sides and diagonals of the quadrilateral.
3. How can we apply the Mid-Point Theorem in solving problems related to quadrilaterals?
Ans. To apply the Mid-Point Theorem in solving problems related to quadrilaterals, we need to identify the midpoints of the sides of the quadrilateral. By drawing the line segments connecting these midpoints, we can determine if they are parallel or not. This information can then be used to solve various geometrical problems and prove other properties of the quadrilateral.
4. Can the Mid-Point Theorem be used to prove that a quadrilateral is a parallelogram?
Ans. Yes, the Mid-Point Theorem can be used to prove that a quadrilateral is a parallelogram. If the line segment joining the midpoints of one pair of opposite sides is parallel to the line segment joining the midpoints of the other pair of opposite sides, then the quadrilateral is a parallelogram. This is a direct consequence of the Mid-Point Theorem.
5. What are some real-life applications of the Mid-Point Theorem in quadrilaterals?
Ans. The Mid-Point Theorem in quadrilaterals has various real-life applications. It can be used in architecture and construction to ensure the parallelism of certain lines or to divide a line segment into equal parts. It is also used in computer graphics to create visually appealing and accurate representations of quadrilateral shapes. Additionally, the Mid-Point Theorem is applied in engineering and design to optimize the placement of objects and determine optimal dimensions.
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