Problem 15 - Thinking Mathematically

Question

In Exercises 15-20, find the measure of the complement and the supplement of each angle. \(48^{\circ}\)

Step-by-Step Solution

Answer

The complement of \(48^{\circ}\) is \(42^{\circ}\) and the supplement is \(132^{\circ}\).

Step 1: Finding the Complement

To find the complementary angle of \(48^{\circ}\), subtract the given angle from \(90^{\circ}\). This can be formulated as \(90^{\circ} - 48^{\circ}\).

Step 2: Finding the Supplement

To find the supplementary angle of \(48^{\circ}\), subtract the given angle from \(180^{\circ}\). This can be formulated as \(180^{\circ} - 48^{\circ}\).

Key Concepts

Complement of an Angle

Angles are fundamental to understanding geometry, and the complement of an angle is an introductory concept that every student should grasp. In the world of angles, two angles are said to be complementary if their measures add up to exactly $90^\{circ\}$. This is primarily relevant for acute angles, which are angles measuring less than $90^\{circ\}$.

For instance, if you a have a small slice of a pizza that forms a $48^\{circ\}$ angle and you want to find out how big the rest of the pizza slice needs to be to complete a right angle, you're looking for the complement of $48^\{circ\}$. Simply put, the complementary angle can be found by subtracting your angle from $90^\{circ\}$, which in this example gives us: $90^\{circ\}\; -\; 48^\{circ\}\; =\; 42^\{circ\}$.

Supplement of an Angle

Moving beyond complementary angles, you'll encounter the notion of the supplement of an angle. Two angles are supplementary if they add up to a straight line, which is $180^\{circ\}$. Typically, these will be either two acute angles, two obtuse angles, or one of each that align to create a straight angle.

Imagine drawing a straight line and then placing another line just off center so it creates an angle with the first line. If this angle is $48^\{circ\}$, you might wonder what the measure of the other angle formed on the opposite side is. To find this supplementary angle, subtract $48^\{circ\}$ from $180^\{circ\}$, yielding the answer: $180^\{circ\}\; -\; 48^\{circ\}\; =\; 132^\{circ\}$. This is how we determine the size of that other angle to 'complete' the straight line.

Angle Measurement

Understanding angle measurement is essential to studying geometry. Angles are measured in degrees, symbolized by a small superscript circle ($^\{circ\}$). The concept of degree as a unit of angular measure divides a full circle into 360 degrees. From here, we categorize angles by their size:

• Acute angles are less than $90^\{circ\}$
• Right angles are exactly $90^\{circ\}$
• Obtuse angles are between $90^\{circ\}$ and $180^\{circ\}$
• Straight angles are $180^\{circ\}$
• Reflex angles are between $180^\{circ\}$ and $360^\{circ\}$

Protractors and Angle Measurement

A protractor, a semicircular or circular tool with degree markings, is used to measure the size of an angle. Position the center of the protractor at the vertex of the angle, and align one ray with the zero mark. The degree reading on the other ray then provides the measure of the angle.

Geometry

Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher-dimensional analogs. It is instrumental in fields such as architecture, engineering, and even art.

An intuitive way to appreciate geometry starts with understanding simple shapes and how they relate to one another. We begin with the point, move to the line, progress to shapes like triangles, rectangles, and circles, and from there, learn about more complex structures.
Geometry also explores the properties and relations of spaces. Concepts like angles, the complement and supplement, are basics that build up to more advanced subjects like trigonometry and calculus.

Real-life Applications of Geometry

Geometry affects our lives daily, from the design of our cities to the animation in video games. By mastering the fundamentals such as complements and supplements of angles, students can understand real-world structures more deeply and develop crucial problem-solving skills.