Problem 42 - Precalculus : Building Concepts and Connections

Question

Find the angle that is complementary to it. $$33^{\circ}$$

Step-by-Step Solution

Answer

The unknown angle is 57 degrees.

Step 1: Understand the Concept of Complementary Angles

Complementary angles are two angles that add up to 90 degrees. If one angle is known, the other can be found by subtracting the known angle from 90 degrees.

Step 2: Apply the Knowledge of Complementary Angles to Find the Unknown Angle

Let's denote the unknown angle as \(x\). Since the two angles are complementary, we have \(x + 33 = 90\). Therefore, to find the unknown angle \(x\), we need to subtract 33 degrees from 90 degrees.

Step 3: Calculation

Perform the subtraction \(x = 90 - 33\).

Key Concepts

Angle Subtraction

Understanding angle subtraction is a fundamental part of working with complementary angles. In basic geometry, it is common to find an unknown angle by subtracting from a known reference angle. When dealing with complementary angles, this reference angle is always 90 degrees because complementary angles add up to 90 degrees. When tasked with finding an unknown angle, if the measure of one angle is given, like 33 degrees, the complementary angle can be found by subtracting this known angle from 90 degrees. This subtraction gives us: \[ x = 90^{\circ} - 33^{\circ} = 57^{\circ} \]Where \(x\) represents the unknown complementary angle. By mastering this simple subtraction, you can easily find any angle that, when added to a given angle, equals 90 degrees. Always remember:

• Subtract the given angle from 90 degrees.
• The result is the measure of the unknown complementary angle.

Angle Measurement

The measurement of angles is a crucial concept in geometry. Angles are typically measured in degrees, which is a standard unit of measuring angles. An angle represents the rotation needed to bring one arm of an angle to the other, with the vertex being the common endpoint. The notation \(^{\circ}\) represents degrees, and different angles have specific ranges:

• A right angle is 90 degrees.
• An acute angle is less than 90 degrees.
• An obtuse angle is greater than 90 degrees but less than 180 degrees.

Understanding how to measure angles and express them in degrees is essential for solving problems involving complementary angles. By adding two angles together, if their sum equals 90 degrees, they are complementary. Observing that one angle is 33 degrees, for instance, quickly tells you that its complement is 57 degrees, as the sum of 33 and 57 is 90 degrees. Keep in mind:

• Always express angles in terms of degrees in simple geometry problems.
• Complementary angles always sum to 90 degrees.

Basic Geometry Concepts

Basic geometry includes understanding different angle types and their relationships. One of these fundamental relationships is that of complementary angles, which are pairs of angles that sum up to one right angle, or 90 degrees. Knowing the basic principles of angles helps build a strong foundation for more complex geometry problems. Complementary angles provide insight into symmetry and aid in understanding perpendicular lines, which meet at a 90-degree angle. Basic geometry principles to remember:

• A complementary angle must add up to 90 degrees with its paired angle.
• Use subtraction from 90 degrees to find an unknown complementary angle.
• Basic angle understanding helps with solving puzzles and real-world geometric problems.

Grasping these basic geometry concepts is essential. They not only simplify solving textbook problems but also aid in visualizing and understanding the layout of many geometric configurations and real-life applications.