Introduction
A quadrilateral is a polygon with four sides, four angles, and four vertices. Quadrilaterals are all around us — tiles, books, doors, and windows are shaped like quadrilaterals. This chapter explores their properties and the relationships between different types.
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Key Concepts
Angle sum property: The sum of all interior angles of any quadrilateral = 360 degrees.
- Types of Quadrilaterals:
- Trapezium: Exactly one pair of parallel sides.
- Parallelogram: Both pairs of opposite sides are parallel and equal; opposite angles are equal; diagonals bisect each other.
- Rectangle: A parallelogram with all angles = 90 degrees; diagonals are equal.
- Rhombus: A parallelogram with all sides equal; diagonals bisect each other at right angles.
- Square: All sides equal AND all angles 90 degrees; diagonals are equal and bisect each other at right angles.
- Kite: Two pairs of consecutive sides are equal; one pair of opposite angles is equal.
- 1.Properties of Parallelogram:
- 2.Opposite sides are equal and parallel.
- 3.Opposite angles are equal.
- 4.Adjacent angles are supplementary (sum = 180 degrees).
- 5.Diagonals bisect each other.
- Diagonal properties:
- Rectangle: diagonals are equal.
- Rhombus: diagonals bisect each other at 90 degrees.
- Square: both of the above.
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Worked Examples
The angles of a quadrilateral are in ratio 2:3:5:8. Find each angle.
Sum = 360 degrees. Let angles be 2x, 3x, 5x, 8x.
18x = 360 degrees, so x = 20 degrees.
Angles: 40 degrees, 60 degrees, 100 degrees, 160 degrees.
In a parallelogram ABCD, angle A = 70 degrees. Find all angles.
Angle C = 70 degrees (opposite angles equal).
Angle B = 180 - 70 = 110 degrees (adjacent angles supplementary).
Angle D = 110 degrees.
Prove that diagonals of a rectangle are equal.
In rectangle ABCD, triangles ABC and DCB share BC, and AB = DC (opposite sides). Angle ABC = angle DCB = 90 degrees. So triangles are congruent by SAS, giving AC = DB.
The diagonal of a rhombus are 16 cm and 12 cm. Find its side.
Half-diagonals: 8 cm and 6 cm. Side = √(82 + 62) = √(64 + 36) = √(100) = 10 cm.
ABCD is a parallelogram. Diagonals AC and BD meet at O. If AO = 4 cm and BO = 3 cm, find AC and BD.
Since diagonals bisect each other: AC = 2 x AO = 8 cm; BD = 2 x BO = 6 cm.
Three angles of a quadrilateral are 75 degrees, 90 degrees, and 110 degrees. Find the fourth.
Fourth angle = 360 - (75 + 90 + 110) = 360 - 275 = 85 degrees.
Show that a square is a special rhombus.
A rhombus has all four sides equal. A square also has all four sides equal. Additionally, all angles of a square are 90 degrees. So every square is a rhombus, but with the extra condition of right angles.
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Common mistakes
- Assuming all parallelograms are rectangles — they are not unless angles are 90 degrees.
- Forgetting that the diagonals of a rhombus are NOT equal (only a square has equal diagonals among rhombuses).
- Using 180 degrees instead of 360 degrees for the angle sum of a quadrilateral.
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Summary
Quadrilaterals have an angle sum of 360 degrees. Parallelograms are a versatile family with many special members: rectangles, rhombuses, and squares each add extra properties. Understanding how they are related helps in recognizing which properties apply in a given problem.