| --- |
| 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). |
