Geometry: Types Of Triangles

• classify triangles by angles: Right Triangles, Acute Triangles, Obtuse Triangles, Oblique Triangles
• classify triangles by length of sides: Equilateral Triangles, Isosceles Triangles, Scalene Triangles
• solve some problems involving angles and sides of triangles

Triangles

This lesson reviews the common types of triangles in geometry.

Triangles are three-sided shapes that lie in one plane.
Triangles are polygons that have three sides, three vertices and three angles.
The sum of all the angles in any triangle is 180º.

Classifying Triangles

Triangles can be classified according to the size of its angles. Some examples are right triangles, acute triangles and obtuse triangles.

The lengths of the sides of triangles is another common classification for types of triangles. Some examples are equilateral triangles, isosceles triangles and scalene triangles.

Triangles classified by their angles

Right Triangles

A right triangle is a triangle with a right angle (i.e. 90°).

You may have noticed that the side opposite the right angle is always the triangle's longest side. It is called the hypotenuse of the triangle. The other two sides are called the legs. The lengths of the sides of a right triangle are related by the Pythagorean Theorem. There are also special right triangles.

Example 1: A right triangle has one other angle that is 35º. What is the size of the third angle?

Solution:

Step 1:A right triangle has one angle = 90°. Sum of known angles is 90° + 35º = 125°.

Step 2:The sum of all the angles in any triangle is 180º. Subtract sum of known angles from 180°. 180° – 125° = 55°

Answer:The size of the third angle is 55°

Acute Triangles

An acute triangle is a triangle whose angles are all acute (i.e. less than 90°). In the acute triangle shown below, a, b and c are all acute angles.

Example 1: A triangle has angles 46º, 63º and 71º. What type of triangle is this?

Answer: Since all its angles are less than 90°, it is an acute triangle.

Obtuse Triangles

An obtuse triangle has one obtuse angle (i.e. greater than 90º). The longest side is always opposite the obtuse angle. In the obtuse triangle shown below, a is the obtuse angle.

Example 1: Is it possible for a triangle to have more than one obtuse angle?

Solution:

Step 1: Let the angles of the triangle be a, b and c. Let a be the obtuse angle.

Step 2: The sum of all the angles in any triangle is 180º.  a + b + c = 180º.

If a > 90º then b + c must be less than 90º.  Therefore, b and c must be acute angles.

Answer: No, a triangle can only have one obtuse angle.

Oblique Triangles

An oblique triangle is any triangle that is NOT a right triangle. Acute triangles and obtuse triangles are oblique triangles.

Triangles Classified by their sides

Besides classifying types of triangles according to the size of its angles as above: right triangles, acute triangles and obtuse triangles; types of riangles can also be classified according to the length of its sides. Some examples are equilateral triangles, isosceles triangles and scalene triangles.

Equilateral Triangles

An equilateral triangle has all three sides equal in length. Its three angles are also equal and they are each 60º. An equilateral triangle is also an equiangular triangle since all its angles are equal.

Example 1: An equilateral triangle has one side that measures 5 in. What is the size of the angle opposite that side?

Solution:

Step 1: Since it is an equilateral triangle all its angles would be 60º. The size of the angle does not depend on the length of the side.

Answer: The size of the angle is 60º.

Isosceles Triangles

An isosceles triangle has two sides of equal length. The angles opposite the equal sides are also equal.

Example 1: An isosceles triangle has one angle of 96º. What are the sizes of the other two angles?

Solution:

Step 1: Since it is an isosceles triangle it will have two equal angles. The given 96º angle cannot be one of the equal pair because a triangle cannot have two obtuse angles. (Refer to obtuse triangle above).

Step 2: Let x be one of the two equal angles. The sum of all the angles in any triangle is 180°. 
      x + x + 96° = 180° Þ 2x = 84° Þ x = 42°

Answer: The sizes of the other two angles are 42º each.

Example 2: A right triangle has one other angle that is 45º. Besides being right triangle what type of triangle is this?

Solution:

Step 1: Since it is right triangle it will have one 90º angle. The other angle is given as 45º.

Step 2: Let x be third angle. The sum of all the angles in any triangle is 180º.
      x + 90º + 45º = 180° Þ x = 45º

Step 3: Two of the angles are equal which means that it is an isosceles triangle.

Answer: It is also an isosceles triangle.

Scalene Triangles

A scalene triangle has no sides of equal length. Its angles are also all different in size.

Quick Recap On Types Of Triangles

We've looked at classifying types of triangles according to the size of its angles: right triangles, acute triangles and obtuse triangles; and also covered types of triangles according to the length of its sides: equilateral triangles, isosceles triangles and scalene triangles.

We can also name triangles according to both their angles and sides.

Acute isosceles triangles, right isosceles triangles, obtuse isosceles triangles
Acute scalene triangles, right scalene triangles, obtuse scalene triangles

Videos

Types of Triangles - Equilateral, Isosceles, Scalene

Triangle classifications - scalene, isosceles, equilateral, acute, obtuse, right, and equiangular

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