Question

The sum of the measures of the two acute angles in a right triangle is _____. (a) \(45^{\circ}\) (b) \(90^{\circ}\) (c) \(180^{\circ}\) (d) \(360^{\circ}\)

Short answer

The sum of the two acute angles in a right triangle is 90 degrees.

Step-by-step solution

- Understand Triangle Angle Sum

A triangle's interior angles add up to 180 degrees. This is true for all types of triangles, including right triangles.

- Identify Angles in a Right Triangle

A right triangle has one angle that measures 90 degrees. The other two angles must be acute angles.

- Set Up the Equation

Since the total sum of the angles in any triangle is 180 degrees, and one angle is 90 degrees, the equation for the remaining two acute angles is: \[\text{Sum of the two acute angles} = 180^{\circ} - 90^{\circ}\]

- Solve the Equation

Subtract 90 degrees from 180 degrees: \[180^{\circ} - 90^{\circ} = 90^{\circ}\]

- Select the Correct Answer

The sum of the measures of the two acute angles in a right triangle is 90 degrees. Therefore, the correct answer is (b) 90 degrees.

Key concepts

sum of angles in a triangle

In any triangle, the sum of all interior angles is always 180 degrees. This is a fundamental property of triangles in Euclidean geometry. No matter if the triangle is acute, obtuse, equilateral, or right-angled, this rule holds true.
This means when you add up the three angles inside any triangle, you will get 180 degrees. This property helps to solve various problems and understand different types of triangles.
For example, if you know two angles in a triangle, you can easily find the third angle by subtracting the sum of the known angles from 180 degrees.
Understanding this concept is crucial as it forms the basis for solving many geometry problems. Especially, in right triangles, this property is heavily used.

right triangle properties

A right triangle has several distinct properties that make it unique. The most notable is that one of its interior angles measures exactly 90 degrees, also called a right angle.
The presence of this right angle shapes the other properties of the triangle:

• The side opposite the right angle is the longest side, known as the hypotenuse.
• The other two sides are called the legs of the triangle.
• The remaining two angles in the triangle must be acute since their sum must be 90 degrees to meet the total 180 degrees rule for triangles.

These properties are useful in many practical applications, such as construction, trigonometry, and more. Knowing these can help you identify when to use specific formulas and rules.

acute angles in a triangle

In any right triangle, the two angles that are not the right angle are called acute angles. An acute angle is one that measures less than 90 degrees.
Because the sum of all the angles in a triangle is 180 degrees, and one of these angles is already 90 degrees (the right angle), the sum of the other two angles must be 90 degrees.
This ensures that each of these two remaining angles is less than 90 degrees, categorizing them as acute.
This property is very helpful because it simplifies finding unknown angles in right triangles. These acute angles will always add up to 90 degrees, allowing for straightforward calculations and confirming that we are dealing with a right triangle.