Problem 49 - Precalculus : Building Concepts and Connections

Question

Find the angle that is supplementary to it. $$130^{\circ}$$

Step-by-Step Solution

Answer

The angle that is supplementary to 130 degrees is \( 180^\circ - 130^\circ = 50^\circ \). So, the supplementary angle is 50 degrees.

Step 1: Understanding Supplementary Angles

Supplementary angles are two angles that add up to 180 degrees. So, if we are given one angle, the supplementary angle would be the difference between 180 degrees and the given angle.

Step 2: Find the supplementary angle to 130 degrees

To find out the supplementary angle to 130 degrees, subtract 130 degrees from 180 degrees. Mathematically, this is shown as: \[180^\circ - 130^\circ\]

Key Concepts

Angles in Geometry

Geometry is rich with various types of angles that have unique relationships with each other. At its core, an angle in geometry represents the figure formed by two rays, called the sides of the angle, sharing a common endpoint, known as the vertex. The magnitude of an angle is measured by the amount of rotation one ray must make around the vertex to coincide with the other ray. This measurement is most commonly expressed in degrees. For instance, when there is no rotation and the rays lie on top of one another, the angle is 0 degrees, while a full rotation forms an angle of 360 degrees. Important types of angles based on their measure include:

• Acute Angles: Angles that are less than 90 degrees.
• Right Angles: Angles exactly equal to 90 degrees.
• Obtuse Angles: Angles greater than 90 degrees but less than 180 degrees.
• Straight Angles: Angles that are exactly 180 degrees, resembling a straight line.

Understanding the basic concept of angles is crucial when it comes to studying their relationships, such as with supplementary angles, an important aspect in solving many geometric problems.

Degrees in Angles

Degrees are the unit of measurement for angles in geometry and they provide a way for us to quantify the size of an angle. The concept of degrees dates back to ancient civilizations, which divided the circle into 360 parts, called degrees. A degree, denoted by the symbol \( ^\circ \), represents \( \frac{1}{360} \) of a full rotation around a circle, making it a useful unit for measuring both small and large angles. For example, a quarter turn (a right angle) is 90 degrees, and a half turn (a straight angle) is 180 degrees. The system of degrees allows mathematicians and students alike to communicate and work with angles in a clear, universal, and practical manner. It's also critical to understand this system when dealing with angle relationships, such as complementary (angles that add up to 90 degrees) and supplementary angles.

Complementary and Supplementary Angles

Geometry introduces us to specific angle pairs that have special names due to their relationships. Complementary angles are two angles whose measures add up to 90 degrees, often found in right angle triangles, while supplementary angles are two angles whose measures sum up to 180 degrees, as commonly seen at a straight line.
The understanding of supplementary angles is essential when solving various geometry problems. For example, when one angle is given, such as 130 degrees in an exercise, its supplementary angle can be found by subtracting the given angle from 180 degrees, as supplementary angles always add to a straight angle. Mathematically, it is represented as: \[ 180^\circ - \text{given angle} \]. In the given exercise, the supplementary angle would be \[ 180^\circ - 130^\circ = 50^\circ \]. This particular understanding of angles not only helps in geometry problems but also enriches logical reasoning and spatial awareness which are valuable skills in various applications of mathematics.