Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the measure of the supplement of each angle. $$90^{\circ}$$

Step-by-Step Solution

Answer

The supplementary angle to \(90^{\circ}\) is \(90^{\circ}\)

Step 1 - Understand Supplementary Angles

Understand that supplementary angles are two angles whose measures add up to \(180^{\circ}\).

Step 2 - Find the Supplement

To find the supplement of the given angle, subtract the measure of the given angle from \(180^{\circ}\). In this case, subtract \(90^{\circ}\) from \(180^{\circ}\).

Key Concepts

Understanding Angle Measurement

Angles are a fundamental part of geometry, helping us describe rotations and turns. An angle is created when two rays meet at a common endpoint called a vertex. The amount of rotation between these two rays is measured in degrees, a unit that reflects the extent of turn.
An entire rotation around a point is equal to 360 degrees. So when we say an angle is 90 degrees, it’s a right angle, representing one-quarter of a full circle. In our exercise, understanding the precise measurement of angles is crucial because we need to find a supplementary angle. Supplementary angles together total 180 degrees, or half a circle's rotation.

Basics of Geometry and Angles

Geometry is the branch of mathematics dealing with shapes, sizes, and properties of space. Within this framework, angles play a critical role in defining various shapes and their characteristics.
Angled shapes like triangles, squares, and polygons depend on the precise measurement of their angles to determine their form and properties.
For instance, a square has four equal angles of 90 degrees each. When dealing with supplementary angles, we are often concerned with two angles on a straight line. Here, the two angles together will sum to 180 degrees, demonstrating their supplementary relationship. Recognizing such relationships in geometry helps solve numerous problems and understand the spatial structure.

Using Subtraction to Find Supplementary Angles

Subtraction is the key technique used to find a supplementary angle when you already have one angle’s measure.
To discover the measure of an angle’s supplement, start with 180 degrees—since supplementary angles always add up to this—and subtract the known angle.
In this exercise, with a given angle of 90 degrees, the complementary subtraction is straightforward: 180 degrees minus 90 degrees equals 90 degrees.
The subtraction operation here finds the missing piece needed to complete the supplementary angle pair, which offers insight into a broader range of problems within both math and daily applications. Being comfortable with subtraction in such contexts is vital for anyone working frequently with angles and geometric calculations.