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).