Problem 63 - Algebra for College Students

Question

Solve each problem by using a system of equations. The measure of the larger of two complementary angles is \(15^{\circ}\) more than four times the measure of the smaller angle. Find the measures of both angles.

Step-by-Step Solution

Answer

The smaller angle is \(15^{\circ}\) and the larger angle is \(75^{\circ}\).

Step 1: Defining Variables

Let's define the variables to represent the measures of the angles. Let \( x \) be the smaller angle, and \( y \) be the larger angle. Complementary angles add up to \( 90^{\circ} \), so we have \( x + y = 90^{\circ} \).

Step 2: Setting Up the Equation for the Larger Angle

According to the problem statement, the larger angle is \( 15^{\circ} \) more than four times the smaller angle. This gives us the equation \( y = 4x + 15^{\circ} \).

Step 3: Solving the System of Equations

We have the system of equations:1. \( x + y = 90^{\circ} \)2. \( y = 4x + 15^{\circ} \)Substitute \( y = 4x + 15^{\circ} \) from the second equation into the first equation:\[ x + (4x + 15^{\circ}) = 90^{\circ} \] Simplifying, we get:\[ 5x + 15^{\circ} = 90^{\circ} \]Subtract \( 15^{\circ} \) from both sides:\[ 5x = 75^{\circ} \] Divide by 5:\[ x = 15^{\circ} \]

Step 4: Finding the Larger Angle

Use the value of \( x \) to find \( y \) using the second equation \( y = 4x + 15^{\circ} \):\[ y = 4(15^{\circ}) + 15^{\circ} = 60^{\circ} + 15^{\circ} = 75^{\circ} \]

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This concept is similar to how friends can complement each other, making a whole, like a complete team. In geometry, complementary angles are often used to solve various problems involving angle measures.
For instance, if you know one angle in a complementary pair, you can easily find the other angle by subtracting the measure of the known angle from 90 degrees.

• If angle A is 40 degrees, then its complement, angle B, is 50 degrees, because 40 + 50 equals 90.
• Another example is if angle X is 30 degrees, its complement, angle Y, would be 60 degrees.

This basic understanding is crucial when dealing with word problems involving complementary angles.

Angle Measures

The measure of an angle tells us how much it opens, typically expressed in degrees. A full circle is 360 degrees, and a right angle is 90 degrees. Understanding angle measures helps you calculate the exact values of angles, especially when they form part of more complex shapes.
In problems dealing with complementary angles, like in our exercise, knowing one angle allows you to determine the other angle by the relationship that their measures must sum to 90 degrees.

Here’s a simple tip:

• If you have a special relationship between angles, like in complementary angles, make sure you use it to find the necessary angle measures easily.

In our specific problem, we had to determine the larger angle based on its described relationship to the smaller angle, which involved using the properties of complementary angles.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than 1. In the context of angle problems, these equations are often used to describe relationships between different angle measures.
For example, in our exercise, we established two linear equations based on the problem description:

• The first equation is from understanding complementary angles:
  \(x + y = 90^{\circ} \), which says the measures of the two angles add up to 90 degrees.
• The second equation came from identifying the relationship of the larger angle to the smaller:
  \(y = 4x + 15^{\circ} \), showing that the larger angle is 15 degrees more than four times the smaller angle.

Solving these equations simultaneously gives the measures of both angles. This fundamental mathematical tool is incredibly powerful for solving a variety of problems beyond geometry.