Understanding Complement and Supplement

Oct 13, 2025 by miodeya.com 23 views

Problem Explanation

The problem involves an angle and its complement and supplement. We are given that four times the measure of the complement of an angle is equal to $\frac{2}{3}$ the measure of the original angle's supplement. We need to find the measure of the angle.

Understanding Complement and Supplement

The complement of an angle is the angle that, when added to the given angle, forms a right angle (90 degrees). The supplement of an angle is the angle that, when added to the given angle, forms a straight line (180 degrees).

Let's Denote the Angle as x

Let's denote the measure of the angle as x. Then, the measure of its complement is 90 - x, and the measure of its supplement is 180 - x.

Formulating the Equation

According to the problem, four times the measure of the complement of an angle is equal to $\frac{2}{3}$ the measure of the original angle's supplement. We can write this as an equation:

4(90 - x) = $\frac{2}{3}$ (180 - x)

Simplifying the Equation

Now, let's simplify the equation by distributing the numbers outside the parentheses:

360 - 4x = $\frac{2}{3}$ (180 - x)

Multiplying Both Sides by 3

To get rid of the fraction, let's multiply both sides of the equation by 3:

1080 - 12x = 2(180 - x)

Distributing the 2

Now, let's distribute the 2 on the right-hand side of the equation:

1080 - 12x = 360 - 2x

Adding 12x to Both Sides

Let's add 12x to both sides of the equation to get all the x terms on one side:

1080 = 360 + 10x

Subtracting 360 from Both Sides

Now, let's subtract 360 from both sides of the equation:

720 = 10x

Dividing Both Sides by 10

Finally, let's divide both sides of the equation by 10 to solve for x:

72 = x

Conclusion

The measure of the angle is 72 degrees.

Answer

The correct answer is C. $72^{\circ}$ .

Discussion

This problem involves using algebraic equations to solve for the measure of an angle. It requires understanding the concepts of complement and supplement of an angle and how to formulate and solve equations. The problem also requires attention to detail and careful simplification of the equation.

Key Concepts

This problem involves the following key concepts:

Complement and supplement of an angle
Algebraic equations
Solving equations for the measure of an angle
Trigonometric functions

Applications

This problem has applications in various fields, including:

Geometry and trigonometry
Algebra and calculus
Physics and engineering
Computer science and programming

Real-World Examples

This problem has real-world examples in various fields, including:

Architecture and design
Engineering and construction
Physics and astronomy
Computer science and programming

Practice Problems

Here are some practice problems that involve using algebraic equations to solve for the measure of an angle:

Find the measure of an angle when its complement and supplement are given.
Use algebraic equations to solve for the measure of an angle when its complement and supplement are related in a certain way.
Solve equations involving trigonometric functions to find the measure of an angle.

Solutions

Here are the solutions to the practice problems:

Find the measure of an angle when its complement and supplement are given: Let's denote the measure of the angle as x. Then, the measure of its complement is 90 - x, and the measure of its supplement is 180 - x. We can write the equation as: 4(90 - x) = $\frac{2}{3}$ (180 - x) Simplifying the equation, we get: 360 - 4x = $\frac{2}{3}$ (180 - x) Multiplying both sides by 3, we get: 1080 - 12x = 2(180 - x) Distributing the 2, we get: 1080 - 12x = 360 - 2x Adding 12x to both sides, we get: 1080 = 360 + 10x Subtracting 360 from both sides, we get: 720 = 10x Dividing both sides by 10, we get: 72 = x Therefore, the measure of the angle is 72 degrees.
Use algebraic equations to solve for the measure of an angle when its complement and supplement are related in a certain way: Let's denote the measure of the angle as x. Then, the measure of its complement is 90 - x, and the measure of its supplement is 180 - x. We can write the equation as: 4(90 - x) = $\frac{2}{3}$ (180 - x) Simplifying the equation, we get: 360 - 4x = $\frac{2}{3}$ (180 - x) Multiplying both sides by 3, we get: 1080 - 12x = 2(180 - x) Distributing the 2, we get: 1080 - 12x = 360 - 2x Adding 12x to both sides, we get: 1080 = 360 + 10x Subtracting 360 from both sides, we get: 720 = 10x Dividing both sides by 10, we get: 72 = x Therefore, the measure of the angle is 72 degrees.
Solve equations involving trigonometric functions to find the measure of an angle: Let's denote the measure of the angle as x. Then, the sine of the angle is sin(x). We can write the equation as: sin(x) = $\frac{1}{2}$ Solving for x, we get: x = arcsin( $\frac{1}{2}$ ) x = 30 degrees Therefore, the measure of the angle is 30 degrees.

Q: What is the complement of an angle?

A: The complement of an angle is the angle that, when added to the given angle, forms a right angle (90 degrees).

Q: What is the supplement of an angle?

A: The supplement of an angle is the angle that, when added to the given angle, forms a straight line (180 degrees).

Q: How do I find the measure of an angle when its complement and supplement are given?

A: To find the measure of an angle when its complement and supplement are given, you can use the following steps:

Let's denote the measure of the angle as x.
The measure of its complement is 90 - x, and the measure of its supplement is 180 - x.
Write the equation as 4(90 - x) = $\frac{2}{3}$ (180 - x)
Simplify the equation by distributing the numbers outside the parentheses.
Multiply both sides by 3 to get rid of the fraction.
Distribute the 2 on the right-hand side of the equation.
Add 12x to both sides to get all the x terms on one side.
Subtract 360 from both sides to isolate the x term.
Divide both sides by 10 to solve for x.

Q: How do I use algebraic equations to solve for the measure of an angle when its complement and supplement are related in a certain way?

A: To use algebraic equations to solve for the measure of an angle when its complement and supplement are related in a certain way, you can follow the same steps as above.

Q: How do I solve equations involving trigonometric functions to find the measure of an angle?

A: To solve equations involving trigonometric functions to find the measure of an angle, you can use the following steps:

Let's denote the measure of the angle as x.
The sine of the angle is sin(x).
Write the equation as sin(x) = $\frac{1}{2}$
Solve for x by using the inverse sine function.
The measure of the angle is x.

Q: What are some real-world examples of using algebraic equations to solve for the measure of an angle?

A: Some real-world examples of using algebraic equations to solve for the measure of an angle include:

Architecture and design: When designing a building, architects need to use algebraic equations to solve for the measure of angles in order to ensure that the building is structurally sound.
Engineering and construction: Engineers and construction workers need to use algebraic equations to solve for the measure of angles in order to ensure that the building or structure is properly aligned and stable.
Physics and astronomy: Physicists and astronomers need to use algebraic equations to solve for the measure of angles in order to understand the behavior of celestial bodies and predict their movements.
Computer science and programming: Computer scientists and programmers need to use algebraic equations to solve for the measure of angles in order to create algorithms and programs that can accurately calculate and manipulate angles.

Q: What are some practice problems that involve using algebraic equations to solve for the measure of an angle?

A: Some practice problems that involve using algebraic equations to solve for the measure of an angle include:

Find the measure of an angle when its complement and supplement are given.
Use algebraic equations to solve for the measure of an angle when its complement and supplement are related in a certain way.
Solve equations involving trigonometric functions to find the measure of an angle.

Q: What are some tips for solving algebraic equations to find the measure of an angle?

A: Some tips for solving algebraic equations to find the measure of an angle include:

Make sure to follow the order of operations (PEMDAS) when simplifying the equation.
Use inverse trigonometric functions to solve for the measure of the angle.
Check your work by plugging the solution back into the original equation.
Use a calculator to check your work and ensure that the solution is accurate.

Q: What are some common mistakes to avoid when solving algebraic equations to find the measure of an angle?

A: Some common mistakes to avoid when solving algebraic equations to find the measure of an angle include:

Not following the order of operations (PEMDAS) when simplifying the equation.
Not using inverse trigonometric functions to solve for the measure of the angle.
Not checking your work by plugging the solution back into the original equation.
Not using a calculator to check your work and ensure that the solution is accurate.