Video: Identifying the Type of an Angle in a Triangle Using the Triangle Inequality Theorem

Video Transcript

In triangle 𝐴𝐵𝐶, 𝐴𝐵 squared is less than 𝐵𝐶 squared plus 𝐴𝐶 squared. What type of angle is 𝐶?

If we consider any triangle 𝐴𝐵𝐶, there are three possibilities for angle 𝐶. The first diagram shows that angle 𝐶 is a right angle. The second diagram shows that angle 𝐶 is an acute angle, as it is less than 90 degrees. The third diagram shows that angle 𝐶 is an obtuse angle, as it is greater than 90 degrees but less than 180 degrees.

We need to consider the relationship between the lengths of sides in triangles and decide which one corresponds to 𝐴𝐵 squared is less than 𝐵𝐶 squared plus 𝐴𝐶 squared. Pythagoras’s theorem states that, in any right-angled triangle, 𝐴 squared plus 𝐵 squared is equal to 𝐶 squared, where 𝐶 is the longest side of the triangle, known as the hypotenuse. This means that, in our first diagram, 𝐴𝐵 squared is equal to 𝐵𝐶 squared plus 𝐴𝐶 squared.

We can therefore rule out right angle for angle 𝐶. When angle 𝐶 gets larger, as in the third diagram, the length 𝐴𝐵 also becomes larger. We can therefore say that when angle 𝐶 is obtuse, 𝐴𝐵 squared is greater than 𝐵𝐶 squared plus 𝐴𝐶 squared. We can therefore also rule out an obtuse angle.

In the second diagram, when angle 𝐶 becomes smaller, the length 𝐴𝐵 also becomes smaller. This means that when angle 𝐶 is acute, 𝐴𝐵 squared is less than 𝐵𝐶 squared plus 𝐴𝐶 squared. This inequality corresponds to the one in the question. We can therefore conclude that angle 𝐶 is acute.