People encounter triangles and quadrilaterals every day in books, machines and fabric designs, to name a few examples. Unless people know the properties of these polygons, they will not be able distinguish them from similar shapes. The properties of both shapes have important academic and practical relevance. For instance, to determine the size of an angle of a triangle, given the size of other angles, you must know the properties of a triangle. Engineers or artists communicating with each other by telephone need to understand one another when one of them requests that the other bend an iron into a triangle or shape it into a quadrilateral.
Both are Polygons
Triangles and quadrilaterals are both polygons -- that is, they are both two-dimensional shapes with straight edges that meet only at their endpoints. Shapes whose sides are nearly straight may resemble triangles and quadrilaterals; therefore, before you call a shape a triangle or quadrilateral, ensure that its sides are straight lines. Similarly, ensure that their edges do not meet anywhere except at the endpoints of the shape. Hence, triangles and quadrilaterals are closed shapes, with no gaps anywhere. Triangles and quadrilaterals have only length and breadth but not depth and cannot therefore have a volume.
A triangle has three sides, which may or may not be equal. The length of the sides of a triangle is one factor mathematicians use to classify triangles. Thus a scalene triangle is one whose sides are of different lengths, while an equilateral triangle is one with equal sides.
A quadrilateral has four sides that also may or may not be equal. Thus, rectangles, squares and parallelograms are all quadrilaterals with different angular and side-length properties.
Two edges of a polygon form an interior angle. The sum of the interior angles of a triangle is always 180 degrees. Therefore, if you know the size of any two interior angles of a triangle, you can always determine the size of the third angle by subtracting the sum of the two from 180 degrees.
On the other hand, the interior angles of a quadrilateral always total 360 degrees. If you know the sizes of three of the four angles, you can determine the size of the fourth angle by deducting the sum of the three from 360 degrees.
The exterior angle of a shape is the angle any two of its edges form outside of it. Exterior angles lie directly opposite interior angles, and the sum of exterior and interior angles is always 360 degrees. Because triangle and quadrilaterals are polygons, the total sum of their exterior angles is always 360 degrees.
If you know the total size of any two exterior angles of a triangle, you can always determine the size of the third angle by subtracting the sum of the two from 360 degrees. The same applies to a quadrilateral if you know the total size of any three exterior angles.