Quadrilaterals and Areas of Parallelograms: Class 9 Study Guide
A parallelogram has special properties that help us prove facts and compare areas.
A quadrilateral has four sides, and its angles add up to 360°. A parallelogram has both pairs of opposite sides parallel. In a parallelogram, opposite sides are equal, opposite angles are equal and the diagonals bisect each other. Each diagonal splits it into two congruent triangles.
A quadrilateral is a parallelogram if both pairs of opposite sides are equal, or if one pair of opposite sides is both equal and parallel, or if its diagonals bisect each other.
Special parallelograms
| Shape | Extra property |
|---|---|
| Rectangle | All angles 90°; diagonals equal |
| Rhombus | All sides equal; diagonals meet at right angles |
| Square | Both of the above |
The mid-point theorem
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half as long. If the third side is 10 cm, the segment is 5 cm. The converse also holds: a line through the mid-point of one side, parallel to another side, bisects the third side.
Mark them with ticks.
Draw the segment.
Parallel and half as long.
In lengths or proofs.
Areas
Parallelograms on the same base and between the same parallel lines have equal area. The area of a parallelogram is base × height. A triangle on the same base and between the same parallels has half the area of the parallelogram. A parallelogram with base 8 cm and height 5 cm has an area of 40 cm², and the triangle on the same base and between the same parallels has an area of 20 cm².
Worked examples and common mistakes
Example: In parallelogram ABCD, ∠A = 70°, so ∠C = 70° and ∠B = ∠D = 110°. Example: The sides of a triangle are 12, 16 and 20 cm. The mid-point segment parallel to the 20 cm side is 10 cm. Common mistakes: using a slanted side instead of the perpendicular height in an area, and saying that all parallelograms have equal diagonals. Only rectangles do.
Check your understanding: model answers
1. 3x = 120, so x = 40. 2. 9 cm. 3. 54 cm².
✨ Key Takeaways
- A quadrilateral's angles add up to 360°.
- A parallelogram's opposite sides and angles are equal and its diagonals bisect each other.
- The mid-point segment is parallel to the third side and half as long.
- Area = base × perpendicular height.
💬 Questions for Discussion & Reflection
- Opposite angles of a parallelogram are 3x and 120°. Find x.
- A triangle's side is 18 cm. How long is the mid-point segment parallel to it?
- Find the area of a parallelogram with base 9 cm and height 6 cm.