Скачать или смотреть 7) Maths Class 9 Chapter 8 Quadrilaterals- Properties Of Parallelogram Theorem 8.6 | Maths Class 9

7) Maths Class 9 Chapter 8 Quadrilaterals- Properties Of Parallelogram Theorem 8.6 | Maths Class 9
Hello students, in this chapter you are going to learn Quadrilaterals.
We will go topic wise
QUADRILATERALS
8.1 Introduction
8.2 Angle Sum Property of a Quadrilateral
8.3 Types of Quadrilaterals
8.4 Properties of a Parallelogram
8.5 Another Condition for a Quadrilateral to be a Parallelogram
8.6 The Mid-point Theorem

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In this chapter, you have studied the following points :
1. Sum of the angles of a quadrilateral is 360°.
2. A diagonal of a parallelogram divides it into two congruent triangles.
3. In a parallelogram,
(i) opposite sides are equal (ii) opposite angles are equal (iii) diagonals bisect each other
4. A quadrilateral is a parallelogram, if
(i) opposite sides are equal or (ii) opposite angles are equal
or (iii) diagonals bisect each other
or (iv)a pair of opposite sides is equal and parallel
5. Diagonals of a rectangle bisect each other and are equal and vice-versa.
6. Diagonals of a rhombus bisect each other at right angles and vice-versa.
7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.
8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.
9. A line through the mid-point of a side of a triangle parallel to another side bisects the third side.
10. The q uadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

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

8.2 Angle Sum Property of a Quadrilateral The sum of the angles of a quadrilateral is 360º.
This can be verified by drawing a diagonal and dividing the quadrilateral into two triangles.

Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent
triangles.
Theorem 8.2 : In a parallelogram, opposite sides are equal.
Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it
is a parallelogram.
Theorem 8.4 : In a parallelogram, opposite angles are equal.
Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then
it is a parallelogram.
Theorem 8.6 : The diagonals of a parallelogram bisect each other
Theorem 8.7 : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
8.5 Another Condition for a Quadrilateral to be a Parallelogram
Theorem 8.8 : A quadrilateral is a parallelogram if a pair of opposite sides is
equal and parallel.
8.6The Mid-point Theorem
Theorem 8.9 : The line segment joining the mid-points of two sides of a triangle
is parallel to the third side.
Theorem 8.10 : The line drawn through the mid-point of one side of a triangle,
parallel to another side bisects the third side.

EXERCISE 8.1
1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the
quadrilateral.
2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
4. Show that the diagonals of a square are equal and bisect each other at right angles.
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
6. Diagonal AC of a parallelogram ABCD bisects
∠ A (see Fig. 8.19). Show that
(i) it bisects ∠ C also, (ii) ABCD is a rhombus.
7. ABCD is a rhombus. Show that diagonal AC
bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.
8. ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:
(i) ABCD is a square (ii) diagonal BD bisects ∠ B as well as ∠ D.