Problem 19 - Thinking Mathematically

Question

Find the measure of the complement and the supplement of each angle. \(37.4^{\circ}\)

Step-by-Step Solution

Answer

The complement of \(37.4^{\circ}\) is \(52.6^{\circ}\) and the supplement of \(37.4^{\circ}\) is \(142.6^{\circ}\).

Step 1: Find the Complement

Firstly, we need to find the complement of the angle. Complementary angles add up to \(90^{\circ}\). So, to find the complement of \(37.4^{\circ}\), subtract \(37.4^{\circ}\) from \(90^{\circ}\) using the formula: \[ Complement = 90 - Angle \] which translates to:\[ Complement = 90 - 37.4 = 52.6^{\circ} \] Therefore, the complement of \(37.4^{\circ}\) is \(52.6^{\circ}\).

Step 2: Find the Supplement

Secondly, we need to find the supplement of the angle. Supplementary angles add up to \(180^{\circ}\). Therefore, to find the supplement of \(37.4^{\circ}\), subtract \(37.4^{\circ}\) from \(180^{\circ}\) using the formula: \[ Supplement = 180 - Angle \] which translates to:\[ Supplement = 180 - 37.4 = 142.6^{\circ} \] Therefore, the supplement of \(37.4^{\circ}\) is \(142.6^{\circ}\).

Key Concepts

Complementary Angles

Complementary angles are a pair of angles that add up to exactly \(90^{\circ}\). This is a key concept in geometry because it helps us understand how different angles work together to form right angles. If you know the measure of one angle, finding its complement is easy. Just subtract the angle from \(90^{\circ}\). For example:

• If an angle is \(37.4^{\circ}\), its complementary angle is calculated as \(90 - 37.4 = 52.6^{\circ}\).

Complementary angles don't need to be adjacent or in the same figure. They simply need to sum up to \(90^{\circ}\). Remember, the presence of complementary angles indicates that if these two angles were placed together, they would form a right angle, which is one of the fundamental elements in geometry.

Supplementary Angles

Supplementary angles are two angles whose measures total \(180^{\circ}\). These types of angles often appear together in geometry, especially in scenarios involving straight lines or linear pairs. To find the supplement of a given angle, use the simple subtraction method: subtract the known angle from \(180^{\circ}\). For instance:

• For an angle of \(37.4^{\circ}\), its supplementary angle is calculated as \(180 - 37.4 = 142.6^{\circ}\).

Geometry utilizes supplementary angles to help define properties of shapes and solve various problems. A common example of supplementary angles is found where two lines intersect, forming adjacent angles that sum to \(180^{\circ}\). Therefore, understanding how to compute supplementary angles is essential for mastering geometric calculations.

Geometry Concepts

Geometry is the branch of mathematics dealing with shapes, sizes, and properties of space. Some crucial concepts in geometry include understanding angles, lines, and shapes. In our focus on angles, complementary and supplementary angles are foundational concepts.

• Complementary Angles: Useful when working with right triangles or when adjusting angles to form \(90^{\circ}\).
• Supplementary Angles: Important in linear equations and in constructing and deconstructing shapes.

In geometry, visualizing these concepts is often as important as the calculations themselves. Sketching diagrams and practicing problems can greatly enhance understanding. For students, working on a variety of problems helps solidify the relationships between these angles and boosts problem-solving skills.