Problem 53 - Prealgebra

Question

Two angles are complementary angles. If one of the angles is \(23^{\circ},\) then solving the equation \(x+23^{\circ}=90^{\circ}\) will give you the other angle. Solve the equation. CAN'T COPY THE GRAPH

Step-by-Step Solution

Answer

The other angle measures \(67^{\circ}\).

Step 1: Understand Complementary Angles

Complementary angles are two angles whose measures add up to \(90^{\circ}\). In this exercise, you are given that one angle measures \(23^{\circ}\) and need to find the measure of the second angle.

Step 2: Write the Equation

Since the two angles are complementary, they sum up to \(90^{\circ}\). Given one angle as \(23^{\circ}\), let \(x\) be the measure of the other angle. The equation is \(x + 23^{\circ} = 90^{\circ}\).

Step 3: Solve for the Unknown Angle

To find \(x\), subtract \(23^{\circ}\) from both sides of the equation: \[ x + 23^{\circ} = 90^{\circ} \]\[ x = 90^{\circ} - 23^{\circ} \]

Step 4: Calculate the Value of \(x\)

Perform the subtraction: \[ x = 90^{\circ} - 23^{\circ} = 67^{\circ} \] This means the measure of the other angle is \(67^{\circ}\).

Key Concepts

Angle Measurement

In the world of geometry, understanding how to measure angles is fundamental. An angle is formed when two lines meet at a point, and its measurement tells us the degree of rotation from one line to the other. The unit of measurement for angles is degrees (`°`).

There are several types of angles, such as acute, right, obtuse, and straight. Complementary angles, specifically, are pairs of angles that add up to exactly `90°`.

• An easy way to remember this is that a right angle is `90°`, and complementary angles together fill up a right angle.
• If you know the measurement of one angle, you can easily find the other by subtracting the known angle from `90°`.

This concept is widely used in various practical and theoretical applications, such as in construction, navigation, and arts. It’s essential to grasp these basics in order to solve more complex problems effectively.

Solving Equations

Equations are the backbone of algebra. They are statements that express the equality of two mathematical expressions. To solve an equation means to find the value of the variable that makes the statement true.

In our specific problem, we are given the equation: \[ x + 23^{\circ} = 90^{\circ} \]Our task is to find the value of `x`.

• First, identify the unknown variable you need to solve for, which is `x`.
• Next, perform operations that isolate `x` on one side of the equation.

This often involves reversing the operations applied to `x`.

Remember, solving equations is all about maintaining equality. Whatever you do to one side of the equation, do the same to the other. Once you simplify all expressions, you'll find the value of the unknown that satisfies the equation.

Basic Subtraction

Subtraction is one of the four foundational arithmetic operations. It represents the operation of removing objects from a collection. In equation solving, subtraction is often used to isolate variables. Let's look at how it applies to the problem of complementary angles.

The equation you are dealing with is:\[ x + 23^{\circ} = 90^{\circ} \]To isolate `x`, you need to subtract `23°` from both sides.

• This operation is like having a balance scale: what you do to one side, you must do to the other to keep it balanced.
• By subtracting `23°` from `90°`, it simplifies to:\[ x = 90^{\circ} - 23^{\circ} \]

Performing this subtraction, we find:\[ x = 67^{\circ} \]Subtraction is straightforward, but fundamental, allowing you to pivot from known quantities to unknowns and complete these calculations.