# Math Accelerated Chapter 11: Congruence, Similarity, and Transformations

## How can we determine if a triangle is a right-angled triangle?

If the measures of the angles of a triangle are in the ratio 4 : 5 : 9, how can we determine if the triangle is a right-angled triangle?

## Answer:

To determine if a triangle is a right-angled triangle, we need to look at the angles within the triangle. In a right-angled triangle, one of the angles is a right angle, which is equal to 90 degrees. This means one of the angles of the triangle should be 90 degrees.

A right triangle, also known as a right-angled triangle, is a type of triangle that contains one angle that measures 90 degrees. This angle is referred to as a right angle and is formed by the intersection of two perpendicular sides. In the context of the given data where the measures of the angles of a triangle are in the ratio 4 : 5 : 9, let's denote the angles as 4x, 5x, and 9x. By adding them together, we get 4x + 5x + 9x = 180°, which simplifies to 18x = 180°. Solving for x, we find x = 10.

Substituting the value of x back into the angles, we get: 4 × 10° = 40°, 5 × 10° = 50°, and 9 × 10° = 90°. Since one of the angles is 90 degrees, we can conclude that the triangle is indeed a right-angled triangle.

Understanding the properties of right-angled triangles is important in geometry and trigonometry, as they form the basis for various mathematical calculations and applications. By recognizing the characteristics of different types of triangles, we can enhance our problem-solving skills and mathematical comprehension.