1. What is a Quadrilateral?
A quadrilateral is a simple closed figure made by joining four line segments in order. In other words, it is a polygon with four sides.
Formally, a quadrilateral is a polygon with four vertices, four sides and four interior angles. We usually name a quadrilateral in order, for example \(ABCD\).
Everyday examples include book covers, computer screens, windows, playing cards, and many floor tiles.
2. Parts of a Quadrilateral
To understand quadrilaterals clearly, you should know their basic parts. These help in reading diagrams and solving questions easily.
2.1. Sides
A quadrilateral has four sides such as \(AB, BC, CD, DA\).
2.2. Vertices
The four corners of a quadrilateral are called vertices, such as \(A, B, C, D\).
2.3. Angles
The four interior angles are formed at the vertices. Example: \(\angle ABC, \angle BCD\).
2.4. Diagonals
The line segments joining opposite vertices are called diagonals. A quadrilateral has two diagonals: \(AC\) and \(BD\).
3. Angle Sum Property of a Quadrilateral
The sum of all interior angles of a quadrilateral is always:
\( \angle A + \angle B + \angle C + \angle D = 360^\circ \)
3.1. Why the Sum is 360°
If you draw a diagonal inside a quadrilateral, it divides it into two triangles. Since each triangle has angle sum \(180^\circ\), the quadrilateral has total angle sum:
\(180^\circ + 180^\circ = 360^\circ\)
3.2. Quick Example
Suppose the angles are \(90^\circ, 85^\circ, 100^\circ\) and \(x\). Then:
\(90 + 85 + 100 + x = 360\)
\(x = 360 - 275 = 85^\circ\)
4. Convex and Concave Quadrilaterals
Quadrilaterals can be classified by the shape of their angles.
4.1. Convex Quadrilateral
All interior angles are less than \(180^\circ\). All diagonals lie inside the figure.
4.2. Concave Quadrilateral
One interior angle is greater than \(180^\circ\). One diagonal lies outside the figure.
5. Naming and Notation
We usually name quadrilaterals by going around the shape in order (clockwise or anti-clockwise). Example: \(ABCD\).
Equal sides are marked using small identical dashes, and equal angles are marked with identical arcs.