Problem 57 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. If the complement of an angle is \(5^{\circ}\) less than one-sixth of its supplement, find the measure of the angle.

Step-by-Step Solution

Answer

The angle measures \(78^{\circ}\).

Step 1: Define the Unknowns

Let the angle be denoted as \(x\). The complement of an angle is \(90^{\circ} - x\), and the supplement of an angle is \(180^{\circ} - x\).

Step 2: Express the Problem Statement Algebraically

According to the problem, the complement of the angle \((90^{\circ} - x)\) is equal to \(5^{\circ}\) less than one-sixth of its supplement \(\left(\frac{1}{6}(180^{\circ} - x) - 5^{\circ}\right)\).

Step 3: Set Up the Equation

We can set up the equation based on step 2: \[90 - x = \frac{1}{6}(180 - x) - 5\]

Step 4: Simplify the Equation

The equation becomes: \[90 - x = 30 - \frac{x}{6} - 5\]Simplify the right-hand side: \[90 - x = 25 - \frac{x}{6}\]

Step 5: Eliminate the Fraction

Multiply every term by 6 to eliminate the fraction: \[6(90 - x) = 6\left(25 - \frac{x}{6}\right)\]This results in: \[540 - 6x = 150 - x\]

Step 6: Solve for \(x\)

Isolate \(x\) by solving the equation:\[540 - 150 = 6x - x\]\[390 = 5x\]Divide both sides by 5:\[x = 78\]

Step 7: Verify the Solution

The angle is \(78^{\circ}\).The complement is \(90 - 78 = 12^{\circ}\).The supplement is \(180 - 78 = 102^{\circ}\).One-sixth of the supplement is \(\frac{1}{6} \times 102 = 17^{\circ}\).\(17^{\circ} - 5^{\circ} = 12^{\circ}\), which matches the complement.

Key Concepts

Complementary Angles

Complementary angles are a pair of angles whose measures add up to precisely 90 degrees. This can happen in various geometric settings, such as when two angles form a right angle with a common side. It’s crucial to remember this specific sum because it often comes into play in different geometry and trigonometry problems.
To understand it better, think of a right angle as a complete 90-degree turn. If you know one angle and need to find its complement, simply subtract the known angle from 90 degrees. For example, if one angle measures 30 degrees, then the complement is 90 - 30 = 60 degrees.

• Helpful rule: Complementary angles always sum to 90 degrees.
• Used heavily in exercises involving right angles.
• Knowing the measure of one angle helps calculate the other easily.

In the context of the exercise, the problem involved finding an unknown angle when related through a complementary relationship. Understanding complementary angles in this way made it easier to formulate the required algebraic expressions.

Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. They are typically found when angles are around a straight line, creating what is known as a linear pair. Like complementary angles, they are fundamental in understanding relationships between angles in various geometrical figures.
To find the supplement of an angle, subtract the given angle from 180 degrees. For instance, if one angle measures 110 degrees, the supplementary angle would be 180 - 110 = 70 degrees.

• Main point: Supplementary angles always sum to 180 degrees.
• Commonly appear in problems with straight lines or linear pairs.
• Finding one angle can immediately lead you to the supplement.

Within the context of the provided exercise, discovering the supplementary angle was key to setting up the algebraic equation. It also aided in defining the relationship involving one-sixth of the supplement.

Equation Solving

Equation solving involves finding the unknown value or values that satisfy an equation. This process can often involve several steps like setting up an equation based on given conditions, simplifying it, and methodically isolating the variable.
To effectively solve an equation:

• Identify and define unknowns clearly.
• Set up the equation based on the problem's given conditions.
• Simplify the equation, if needed, to make it easier to solve.
• Isolate the unknown variable, often by performing inverse operations.
• Always verify the solution by plugging it back into the original equation to ensure correctness.

In the example exercise, solving the equation involved combining our knowledge of complementary and supplementary angles to form an equation. The problem then required simplifying and rearranging this equation to find the unknown angle, verifying the solution afterwards. This highlights the structured approach of breaking down complex problems into smaller, manageable parts, which is crucial in algebraic problem solving.