Question

The measure of one acute angle of a right triangle is 30 . Find the measure of the other acute angle. (Lesson 5-2)

Short answer

The other acute angle measures 60 degrees.

Step-by-step solution

Identify Triangle Type and Properties

The problem mentions a right triangle, which is characterized by having one angle measuring 90 degrees. Additionally, the sum of the angles in any triangle is always 180 degrees.

Calculate the Sum of Remaining Angles

In a right triangle, if one angle is 90 degrees, the sum of the other two angles must be the difference between 180 degrees and 90 degrees. Calculate as follows: The sum of other two angles = 180° - 90° = 90°.

Use Given Angle to Find the Other Angle

We know one acute angle is 30 degrees. To find the measure of the other acute angle, subtract the 30 degrees from the total sum of the two acute angles: Measure of the other acute angle = 90° - 30° = 60°.

Key concepts

Acute Angle

An acute angle is an angle that measures less than 90 degrees. In the world of geometry, an acute angle is found in various types of triangles, especially the right triangle.

Right triangles have exactly one acute angle. The properties of these angles are crucial for solving many geometry problems. To better understand acute angles, remember that they are always sharp-looking, unlike right angles which are perfectly 90 degrees.

• Acute angles are essential in trigonometry.
• In a right triangle, besides the right angle, the other two must be acute.
• Acute angles vary from just above 0 degrees to just below 90 degrees.

Understanding acute angles helps you solve problems that involve determining unknown angles, especially in triangles.

Triangle Sum Property

The triangle sum property is a fundamental rule in geometry that states the sum of the interior angles of a triangle is always 180 degrees. This is a key principle that is extremely helpful when determining unknown angles in a triangle.

In every triangle, whether it is scalene, isosceles, or right, this sum remains constant.

• This rule is applied universally to all triangles.
• It's highly useful when an angle is missing or needs calculation.
• Remembering this property helps with understanding geometric problems better.

In the case of a right triangle, one angle is always 90 degrees, so the remaining two angles have to add up to 90 degrees. This property links closely with both acute angles and the calculations you perform on them.

Angle Calculation

Calculating angles, especially in triangles, is a common mathematical task. The calculation involves determining the unknown angles using known measures and properties like the triangle sum property.

When working with triangles, always start by considering the type of triangle and what you know:

• Identify any given angles.
• Use properties like the 180-degree rule for triangles.
• Subtract known angles from the total to find unknown ones.

In our problem, once knowing that one acute angle is 30 degrees and being in a right triangle setup, we used the remaining angle sum of 90 degrees to find the missing angle. The straightforward method is simply to subtract the given angle from the total of the acute angles.

Thus, in our example, the other acute angle is found by subtracting 30 from 90, giving us 60 degrees.