Problem 216 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.

Step-by-Step Solution

Answer

The smaller angle is 37 degrees, and the larger angle is 143 degrees.

Step 1: Identify the relationship between the angles

Supplementary angles sum up to 180 degrees. Let the measure of the smaller angle be denoted by x degrees. The larger angle then is 180 - x degrees.

Step 2: Create the equation for the larger angle

According to the problem, the larger angle is five degrees less than four times the measure of the smaller angle. This translates to the equation: 180 - x = 4x - 5

Step 3: Solve for x

Start by adding x to both sides of the equation: 180 = 5x - 5. Next, add 5 to both sides of the equation: 185 = 5x. Finally, divide both sides by 5 to find x: x = 37.

Step 4: Find the larger angle

Substitute x = 37 into the expression for the larger angle: 180 - 37 = 143 degrees.

Step 5: Conclusion

The measures of the two angles are 37 degrees and 143 degrees.

Key Concepts

supplementary angles

Supplementary angles are two angles whose measures add up to 180 degrees. This relationship is essential in understanding various geometric problems. Whenever you have two angles that supplement each other, you can use the equation: \( \text{Angle}_1 + \text{Angle}_2 = 180^\circ \).

In the provided exercise, we are told that two angles are supplementary. This means you can immediately conclude their sum is 180 degrees. This is a powerful piece of information because it lets us set up an equation that eventually helps us solve for each angle.

Knowing this concept is crucial not only for solving this particular problem but also for understanding more complex geometrical relationships. So, always remember: supplementary angles add up to 180 degrees.

algebraic equations

When dealing with geometry problems like the one given in the exercise, translating verbal descriptions into algebraic equations is crucial. Here's how we do that:

First, let's establish our variable. We'll call the measure of the smaller angle \( x \). According to the supplementary angles definition, the larger angle can be expressed as \( 180 - x \).

The problem states that the larger angle is five degrees less than four times the smaller angle. This statement translates to the algebraic equation: \( 180 - x = 4x - 5 \).

Solving this equation step-by-step is crucial:

• Combine like terms: Add \( x \) to both sides to get \( 180 = 5x - 5 \).
• Isolate \( x \): Add 5 to both sides to get \( 185 = 5x \).
• Solve for \( x \): Divide by 5, which gives \( x = 37 \).

Being comfortable with setting up and solving algebraic equations can make seemingly complex problems much easier.

angle relationships

Understanding angle relationships is key to solving problems involving angles. In the exercise, we are dealing specifically with supplementary angles, but knowing different types of angle relationships adds to your toolkit.

Here are a few important relationships to remember:

• Complementary angles: Two angles sum up to 90 degrees.
• Adjacent angles: Two angles share a common vertex and side but do not overlap.
• Vertical angles: Opposite angles formed by two intersecting lines; they are always equal.

In our exercise, leveraging the supplementary angle relationship helps to set up our initial equation. But also understanding how these different types of angles interact can help solve a wide range of problems. For instance, knowing that vertical angles are equal can provide alternative ways to approach problems where intersecting lines create multiple angles.

So, take these relationships and use them as part of your problem-solving strategies. They are powerful tools in geometry!