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source: ncert
topic: geometry
class_level: class_9
chapter: ch8
difficulty: intermediate
---
Class 9 | Ch8: Quadrilaterals
ANGLE SUM: Sum of angles of quadrilateral=360°.
PARALLELOGRAM PROPERTIES:
(i) Opposite sides are parallel and equal.
(ii) Opposite angles are equal.
(iii) Consecutive angles are supplementary (sum=180°).
(iv) Diagonals bisect each other.
THEOREMS:
Theorem 1: Diagonal divides parallelogram into two congruent triangles.
Theorem 2: Opposite sides of parallelogram are equal (and converse).
Theorem 3: Opposite angles of parallelogram are equal (and converse).
Theorem 4: Diagonals of parallelogram bisect each other (and converse).
Theorem 5: Parallelograms on same base and between same parallels are equal in area.
MID-POINT THEOREM:
The line joining midpoints of two sides of triangle is parallel to third side and half its length.
If D=midpoint of AB and E=midpoint of AC then DE parallel to BC and DE=BC/2.
CONVERSE: Line through midpoint of one side parallel to another side bisects the third side.
SOLVED EXAMPLES:
Example 1: ABCD is parallelogram. Angle A=70°. Find all angles.
Angle C=70° (opposite). Angle B=Angle D=180-70=110° (supplementary to A).
Example 2: In parallelogram ABCD, diagonals AC and BD meet at O. AO=5cm. Find AC.
AO=OC (diagonals bisect each other). AC=2×AO=10cm.
Example 3: Mid-point theorem — D and E are midpoints of AB and AC. DE=4cm. Find BC.
BC=2×DE=8cm.
COMMON MISTAKES:
Diagonals of parallelogram bisect each other but are NOT equal (unless rectangle).
Diagonals of rectangle are equal but do NOT bisect at 90° (unless square).
Diagonals of rhombus bisect at 90° but are NOT equal (unless square). |