When lines meet and form complete boundaries, they make shapes with different numbers of sides. These shapes are called polygons. The polygon with the fewest number of sides is a triangle. It has three sides and three angles. A quadrilateral is bounded by four sides. There is no limit to the number of sides a polygon may have.
To summarize then, a polygon is a shape formed by the intersection of three or more lines that make a boundary. The points where the lines meet are called vertices; these are the vertices of the polygon. At these vertices, angles are formed. The lines themselves form the edges of the polygon.
A regular polygon is one whose sides and angles are equal.
Congruent shapes are any two shapes whose angles are equal and sides are proportional (in the same ratio).
A simple polygon has no internal edges, and therefore has no edges that intersect themselves. Figures 3.8 and 3.10 show simple polygons.
A complex polygon has internal edges and intersects itself, as in Figure 3.9.
A convex polygon is a simple polygon whose angles are all less than reflex angles. If a polygon is not convex, it is said to be concave.
A triangle is the simplest polygon. It is made up of three sides and has three angles. Using a combination of triangles, every other polygon can be formed. For example, squares are made up of two adjacent triangles. The sum of angles in a triangle is equal to two right angles, and at least two angles in every triangle are acute. There are four types of triangles.
Equilateral triangle-All sides and angles are equal.
Isosceles triangle-Has at least two equal sides and angles.
For this reason, the equilateral triangle is also a form of the isosceles triangle.
Scalene triangle-No sides or angles are equal.
Right triangle-Has one angle that is a right angle, and so the other two angles are complementary angles.
Figure 3.11: Reproduced from Wikipedia under the GNU license
When faced with a simple polygon that has any number of sides, you can determine how many triangles it is composed of by subtracting 2 from the number of sides. The result will be the number of triangles. For example, how many triangles are in a square? A square has four sides, so 4 − 2 = 2, which means two triangles.
A square is composed of two right triangles. We know the total angle sum in a triangle is two right angles, and therefore the total sum in a square must be twice this amount. A square has four equal sides and four right angles.