Question

The measure of one acute angle of a right triangle is 25 . Find the measure of the other acute angle.

Short answer

The other acute angle measures 65 degrees.

Step-by-step solution

Understanding the Problem

A right triangle has one angle that is always 90 degrees since it's a right angle. The other two angles in the triangle must be acute, meaning they are less than 90 degrees.

Using the Triangle Angle Sum Property

The sum of all angles in a triangle is always 180 degrees. Since one of the angles in a right triangle is 90 degrees, the sum of the other two angles (the acute angles) must be 90 degrees.

Subtract the Known Angle from 90

Since we know one of the acute angles is 25 degrees, subtract this from 90 to find the measure of the other acute angle: \[ 90 - 25 = 65 \]

Conclusion

The measure of the other acute angle is 65 degrees, as both acute angles must sum up to 90 degrees in a right triangle.

Key concepts

Understanding Angle Measurement

Angles are a fundamental concept in geometry. The measurement of an angle is determined by the rotation needed to bring one of the angle's sides to meet the other. Angles are typically measured in degrees (°).
- A full circle is 360°. - Half of that, which is a straight line, is 180°.
There are different types of angles depending on their measurements:

• Acute angles measure less than 90°.
• Right angles measure exactly 90°.
• Obtuse angles measure more than 90° but less than 180°.
• Straight angles measure exactly 180°.

Understanding how to measure and classify angles will help solve problems like finding unknown angles in triangles.

Triangle Angle Sum Property

The triangle angle sum property is a key rule in geometry. It states that the sum of the interior angles of any triangle will always be 180°. This property is vital when solving problems involving triangles. For example, if you know two angles in a triangle, you can easily find the third angle by subtracting the known angles from 180°.
Let's break it down with some general steps when solving:

• Determine the type of triangle you have (right, isosceles, equilateral, etc.).
• Use known angle measurements to find unknown angles by applying the angle sum property.
• In a right triangle, if one angle is 90°, the sum of the other two must be 90°.

This property simplifies finding unknown angles, making it easier to tackle various geometric problems, like finding the measure of an acute angle.

Identifying Acute Angles in Geometry

Acute angles play a key role when analyzing triangles, especially right triangles. In geometry, an acute angle is one that measures less than 90°. This characteristic makes them important when discussing right triangles, since two out of three angles in a right triangle must be acute.
When dealing with right triangles:

• One angle is always 90°, which is the right angle.
• The other two angles are acute.

Right triangles often have practical applications, such as in trigonometry or when considering the properties of special triangles. Knowing that the acute angles in a right triangle sum up to 90°, you can find one unknown angle if the other is known by subtracting the known angle from 90°.
Understanding these concepts will help expand your mathematical toolkit as you encounter more complex problems.