Right Triangles, Acute Triangles and Obtuse Triangles

This lesson reviews the common types of triangles in geometry.

Triangles are three-sided shapes that lie in one plane. Triangles are a type of polygons. The sum of all the angles in any triangle is 180º.

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.

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.

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