Problem 48 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. The ratio of the complement of an angle to its supplement is 1 to 4 . Find the measure of the angle.

Step-by-Step Solution

Answer

The angle measures 60 degrees.

Step 1: Understand the problem

We need to find the measure of an angle given that the ratio of its complement to its supplement is 1:4.

Step 2: Define Variables

Let \( x \) be the measure of the angle. The complement of the angle is \( 90 - x \) and the supplement is \( 180 - x \).

Step 3: Set Up the Ratio Equation

The problem states that the ratio of the complement to the supplement is 1:4. We can set this equation as \( \frac{90 - x}{180 - x} = \frac{1}{4} \).

Step 4: Cross Multiply to Eliminate the Fraction

Cross multiply the equation \( \frac{90 - x}{180 - x} = \frac{1}{4} \) to get \( 4(90 - x) = 180 - x \).

Step 5: Expand and Simplify

Expand the equation from Step 4: \( 360 - 4x = 180 - x \).

Step 6: Solve for \( x \)

Rearrange the equation to solve for \( x \): \( 360 - 180 = 4x - x \), which simplifies to \( 180 = 3x \). Therefore, \( x = \frac{180}{3} = 60 \).

Step 7: Conclusion

The measure of the angle is 60 degrees.

Key Concepts

Ratios

Ratios are a way to compare two quantities by expressing how many times one value contains or is contained within the other. For example, a ratio of 1:4 means that for every one unit of the first quantity, there are four units of the second quantity. Ratios can be expressed in different forms like fractions, decimals, or with a colon.
If you have a ratio problem involving algebra, you'll typically set up an equation that involves both portions of the ratio. This equation helps you solve for the unknown quantity, using the relationship given by the ratio.

• Define your variables. For instance, let a variable represent the initial measure of what you're comparing.
• Convert your word problem into a mathematical equation by expressing the known ratio between the two quantities.
• Don't forget to simplify your equation by cross-multiplying if it involves fractions to make your calculations clearer.

Understanding how to manipulate ratios is key to solving many math problems, especially those involving dimensions, angles, and other proportional relationships in geometry.

Angle Properties

Angles represent the space between two intersecting lines or surfaces at or close to the point where they meet. Knowing angle properties helps one to understand and work with different geometrical figures. Here are some fundamental angle properties that are essential:

• A right angle is exactly 90 degrees. Two lines perpendicular to each other form a right angle.
• Vertical angles are equal; these occur when two lines intersect, forming two pairs of opposite angles.
• Angles on a straight line add up to 180 degrees. This is useful when calculating unknown angles around a straight line.

Understanding these properties, along with knowing complementary and supplementary angles, is crucial for solving problems that involve constructing and interpreting mathematical models. Complementing these properties with algebraic methods can significantly simplify complex angle problems.

Complementary and Supplementary Angles

Complementary and supplementary angles are special pairs of angles that have specific relationships based on their measures.
Complementary angles add up to 90 degrees. This means if you know one of the angles, you can find the other by subtracting the known angle from 90. For example, if one angle is 30 degrees, the other angle that complements it is 60 degrees because 90 - 30 = 60.
Supplementary angles, on the other hand, add up to 180 degrees. If you know one angle in the pair, you can determine the other by subtracting the known angle from 180. For instance, if one angle is 70 degrees, the supplementary angle is 110 degrees because 180 - 70 = 110.
These concepts are particularly useful in geometry when solving for unknown angles. In a problem involving a ratio of an angle's complement to its supplement, these definitions help in setting up the correct algebraic equations. Understanding how to find these angles is crucial for tackling many geometry problems involving parallel lines, triangles, and other shapes.