Introduction
Quadrilaterals — four-sided polygons — are the shapes we encounter most in everyday life: rooms, screens, tables, and fields are all rectangles or close to it. This chapter explores the full family of quadrilaterals, from the general parallelogram to the special cases of rectangle, rhombus, square, trapezoid, and kite, along with the theorems that characterise each.
Key Takeaways
- The sum of the interior angles of any quadrilateral is 360°.
- In a parallelogram, opposite sides are equal, opposite angles are equal, and the diagonals bisect each other.
- A rectangle has all four right angles and equal diagonals; a rhombus has all four sides equal and perpendicular diagonals.
- A square is simultaneously a rectangle and a rhombus — equal sides, equal angles, and perpendicular equal diagonals.
- The midsegment of a trapezoid is parallel to the bases and equal to half their sum.
What is a Quadrilateral?
A quadrilateral (or quadrangle) is a rectilinear figure bound by four straight lines called edges or sides. The word quad means four and lateral means sides. A quadrilateral also has four angles defined by the points where the edges meet (called vertices).
Diagonals are the segments joining opposite vertices.
Adjacent angles share a common side. Opposite angles are not adjacent.
On the basis of diagonals, there are two types: - Convex quadrilateral — the diagonals intersect in the interior region - Concave quadrilateral — one diagonal lies in the exterior region
Convex Quadrilaterals
The most common classes of convex quadrilaterals are: Parallelogram, Trapezium, and Kite.
Parallelogram
A quadrilateral whose opposite sides are pairwise parallel is called a parallelogram.
Theorems: 1. In any parallelogram, opposite sides are congruent, opposite angles are congruent, and the sum of angles adjacent to one side is two right angles. - Corollary: If one angle of a parallelogram is right, then all four are right. 2. If in a convex quadrilateral, opposite sides are congruent to each other, or two opposite sides are congruent and parallel, then it is a parallelogram. 3. A quadrilateral is a parallelogram if and only if its diagonals bisect each other. 4. Two angles whose sides are parallel each to each are either equal or supplementary. 5. The opposite sides of a parallelogram are equal. 6. The diagonals of a parallelogram bisect each other. 7. Two parallelograms are congruent if two sides and the included angle of one equal those of the other. 8. Parallelograms on the same base and in the same parallels are equal to one another. 9. Parallelograms on equal bases and in the same parallels are equal to one another. 10. If a parallelogram has the same base as a triangle and is in the same parallels, then the parallelogram is double the triangle.
Rectangle
When all angles of a parallelogram are right angles, it is called a rectangle.
Theorems: - Each of the four angles of a rectangle is a right angle. - The diagonals of a rectangle are of equal length.
Rhombus
If all sides of a parallelogram are equal to each other, it is called a rhombus.
Theorems: - The diagonals of a rhombus are perpendicular to each other. - Conversely, if the diagonals of a parallelogram are perpendicular, then it is a rhombus.
Square
A square is a parallelogram with all sides equal and all angles right angles — a special case of both rectangle and rhombus.
Theorems: - The diagonals of a square are equal and perpendicular to each other. - If the diagonals of a parallelogram are equal and intersect at right angles, then it is a square.
Trapezium and Trapezoid
When no sides of the quadrilateral are parallel, it is called a trapezium. When exactly two sides are mutually parallel, it is called a trapezoid. If the non-parallel sides are equal, it is an isosceles trapezoid. In modern usage, the terms are often used interchangeably.
Theorems: - A trapezoid is isosceles if and only if the base angles are congruent. - A trapezoid is isosceles if and only if the diagonals are congruent. - If a trapezoid is isosceles, the opposite angles are supplementary. - The median (midsegment) of a trapezoid is parallel to each base and equal to half the sum of the lengths of the bases.
Kite (Deltoid)
A quadrilateral is a kite if it has two pairs of equal adjacent sides and unequal opposite sides.
Polygons in General
A portion of a plane bounded by a broken line is called a polygon. Polygons are classified by number of sides:
| Sides | Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
| 12 | Dodecagon |
- Equilateral polygon — all sides equal
- Equiangular polygon — all angles equal
- Regular polygon — both equilateral and equiangular
- Convex polygon — each angle less than a straight angle
- Concave polygon — has an angle greater than a straight angle
Polygon Theorems
- The sum of the interior angles of a polygon equals two right angles taken (n − 2) times, where n is the number of sides.
- Corollary: Each angle of a regular polygon of n sides equals (n − 2)/n right angles.
- The sum of the exterior angles of a polygon, made by producing each side in succession, equals four right angles.
Generalised Mid-Point Theorem
If three or more parallels intercept equal segments on one transversal, they intercept equal segments on every transversal. - Corollary: The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. - Corollary: The line joining the mid-points of the non-parallel sides of a trapezoid is parallel to the bases and equal to half their sum.
Inscribed and Circumscribed Polygons
An inscribed polygon has all vertices on a circle (the circle is circumscribed about the polygon).
A circumscribed polygon has each side tangent to a circle (the circle is inscribed in the polygon).
Mensuration
Mensuration is the part of geometry concerned with ascertaining lengths, areas, and volumes.
- Perimeter — the total length of the boundary of a figure
- Area — the measure of the surface contained inside a figure, measured against the unit square
Glossary
Circle — a closed curve in a plane, all points equidistant from the centre. A radius is a line from centre to circle. A diameter passes through the centre. An arc is a portion of the circle.
Secant — a line intersecting a circle at two points.
Chord — the portion of a secant inside the circle.
Tangent — a line touching a circle at exactly one point.
Conclusion
Quadrilaterals encompass a rich hierarchy of shapes — from the general four-sided figure down to the uniquely determined square — unified by the constant interior angle sum of 360°. The special properties of parallelograms, rectangles, and rhombuses arise from the parallel postulate and the congruence theorems of the previous chapter, demonstrating how theorems build systematically on each other. Their area formulas are among the most practically useful results in all of elementary geometry.
The next chapter broadens the study to all regular polygons and establishes the general area formulas — Regular Polygons and Areas.