### Video Transcript

In triangle π΄π΅πΆ, π΄π΅ squared is less than π΅πΆ squared plus π΄πΆ squared. What type of angle is πΆ?

If we consider any triangle π΄π΅πΆ, there are three possibilities for angle πΆ. The first diagram shows that angle πΆ is a right angle. The second diagram shows that angle πΆ is an acute angle, as it is less than 90 degrees. The third diagram shows that angle πΆ 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 π΄π΅ squared is less than π΅πΆ squared plus π΄πΆ squared. Pythagorasβs theorem states that, in any right-angled triangle, π΄ squared plus π΅ squared is equal to πΆ squared, where πΆ is the longest side of the triangle, known as the hypotenuse. This means that, in our first diagram, π΄π΅ squared is equal to π΅πΆ squared plus π΄πΆ squared.

We can therefore rule out right angle for angle πΆ. When angle πΆ gets larger, as in the third diagram, the length π΄π΅ also becomes larger. We can therefore say that when angle πΆ is obtuse, π΄π΅ squared is greater than π΅πΆ squared plus π΄πΆ squared. We can therefore also rule out an obtuse angle.

In the second diagram, when angle πΆ becomes smaller, the length π΄π΅ also becomes smaller. This means that when angle πΆ is acute, π΄π΅ squared is less than π΅πΆ squared plus π΄πΆ squared. This inequality corresponds to the one in the question. We can therefore conclude that angle πΆ is acute.