What is a triangle?

A triangle is a polygon formed by three vertices and, consequently, three edges. The sum of its interior angles is always equal to (180^º). Triangles have classifications based on either sides or angles. By sides, triangles can be equilateral (all of whose sides and angles are equal), isosceles (where two of its sides are equal), or scalene (all the sides and angles are different). The triangles may be acute if all angles are less than(90º, right if one angle is exactly (90º, and obtuse if one of the angles is greater than (90º) Triangles are very important for geometry. They are widely used in architecture, engineering, trigonometry, and related to many other applications. Because such properties as area and perimeter calculation make them essential in theoretical and practical applications.

Types of Triangles

Equilateral Triangles

The Equilateral Triangles have the following properties, aside from the general properties of all triangles:

• Three straight sides of equal length

• Three angles, each measuring 60°

• Three lines of symmetry

Isosceles Triangles

The Isosceles triangles have the following properties:

• Two sides of equal length

• Two equal angles

• One line of symmetry

• Scalene Triangles

Scalene Triangle

Scalene triangles have the following properties

• No sides of equal length

• No equal angles

• No lines of symmetry

Acute triangles

Acute triangles  All acute angles exist in acute triangles. That is, acute angle is less than 90°. It is possible to have an acute triangle which is also an isosceles triangle – these are called acute isosceles triangles.

Right Triangles 

The Right Triangles (right-angled triangles) have one right angle equal to 90°.  It is possible to have a right isosceles triangle – a triangle with a right angle and two equal sides. Obtuse triangles….

Obtuse triangles

Notice that obtuse triangles have one obtuse angle (angle which is greater than 90°). It is possible to have an obtuse isosceles triangle – a triangle with an obtuse angle and two equal sides.

Triangle Formula

Triangle Formula are given below as,

• Perimeter of a triangle = a + b + c

           Where,

              b is the base of the triangle.

              h is the height of the triangle.

• If only 2 sides and an internal angle is given then the remaining sides and angles can be calculated using the given  formula.

Solved Problems Using Triangle Formula

Problem: Given: c = 8, b = 12, C = 30°

Notice that we have two side lengths and an angle which is not the included angle. Refer back to the special cases described in the Introduction you will see that with this information there is the possibility that we can obtain two distinct triangles with this information.

As before we need a diagram to appreciate what we are doing.

Triangle Formula Example

The diagram above we are given two sides and a non-included angle. Since we have been given two sides and a non-included angle we make use of the sine formula.

Or

Since we are given b, c and C, we make use of this portion of the formula to find angle B.

Now there is a potential complication here because there is another angle with sine equal to 0.75. Specifically, B would equal

In the first case the angles of the triangle are then:

In the second case we have:

The situation is illustrated in the following figure. In order to solve the triangle completely we need to consider the two cases as shown and proceed to find the remaining unknown a.

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