Problem 41 - Precalculus : Building Concepts and Connections

Question

Find the angle that is complementary to it. $$48^{\circ}$$

Step-by-Step Solution

Answer

The complementary angle to \(48^{\circ}\) is \(42^{\circ}\).

Step 1: Understand the Concept of Complementary Angles

Complementary angles are two angles that add up to \(90^{\circ}\). So, if one angle is known, the other can be found by subtracting the known angle from \(90^{\circ}\).

Step 2: Subtraction

Subtract the given angle, \(48^{\circ}\) from \(90^{\circ}\) to find the complementary angle. So it is \(90^{\circ} - 48^{\circ} = 42^{\circ}\).

Key Concepts

Geometry and Complementary Angles

In geometry, understanding the different types of angles and their properties is crucial. Complementary angles are a fundamental concept that comes into play, especially when dealing with right angles and their components. By definition, a pair of angles is considered complementary when the sum of their measures equals 90 degrees (\(90^\circ\)).

Importantly, this concept doesn't require the angles to be adjacent, meaning they don't have to share a common side or vertex. Visualizing this can be helpful; for instance, if you picture a right angle (a 90-degree angle), think of splitting it into two smaller angles. These two angles, no matter how you divide them, as long as their total is 90 degrees, are complementary to each other. Hence, knowing one angle allows us to easily find its complementary pair.

Angle Relationships

Angle relationships are an integral part of geometry, and understanding them can help solve a myriad of problems involving angles. When two angles are complementary, as mentioned before, they have a unique relationship: their measures add up to 90 degrees. However, this is just one type of relationship.

Diving deeper, there are also supplementary angles (which sum up to 180 degrees), vertical angles (which are opposite each other when two lines cross and are always equal), and adjacent angles (which share a common side and vertex). Recognizing these relationships is useful when working with geometric figures, proofs, or when solving for unknown angles. By leveraging these relationships, you can unlock the measurements of unnamed angles when given limited information.

Subtraction of Angles

The subtraction of angles is a straightforward arithmetic operation that we apply within the context of geometry to find an angle's complement or supplement. Subtracting one angle measure from another is essential when trying to find the unknown measure of an angle that pairs with a known measure to fulfill a specific relationship.

For instance, as demonstrated in the step-by-step solution, to find the complementary angle to \(48^\circ\), we subtract it from 90 degrees: \(90^\circ - 48^\circ = 42^\circ\). This arithmetic action becomes a powerful tool when integrated with the understanding of angle relationships. By mastering these simple calculations, larger concepts in geometry, like finding unknown angles in complex diagrams, become much more approachable.