Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the five-step problem-solving strategy to find the measure of the angle described. The angle's measure is \(60^{\circ}\) more than that of its complement.

Step-by-Step Solution

Answer

The measure of the angle is \(75^{\circ}\).

Step 1: Understand concept of complementary angles

Complementary angles are two angles whose measures add up to \(90^{\circ}\). If one angle is \(x^{\circ}\), its complement is \(90 - x^{\circ}\).

Step 2: Formulate equation based on problem

The problem states the angle's measure is '\(60^{\circ}\) more than that of its complement'. Therefore, the equation to represent this relationship is \(x = (90 - x) + 60\). In other words, the unknown angle \(x^{\circ}\) equals its complement plus \(60^{\circ}\).

Step 3: Simplify and solve the equation

First, simplify the right side of the equation to get \(x = 150 - x\). Then, solving this equation for \(x\), add \(x\) to both sides to get \(2x = 150\). Dividing both sides by 2 to isolate \(x\), gives \(x = 75\).

Step 4: Verify solution

To confirm the solution, insert \(x = 75^{\circ}\) into the equation from step 2 to verify that \(75 = 90 - 75 + 60\). Simplifying the right hand side gives \(75 = 15 + 60\), which is \(75 = 75\), confirming the solution is correct.

Key Concepts

Angle Measurement

Understanding angle measurement is key when dealing with various geometric problems. Angles are typically measured in degrees and serve as a way to quantify the amount of rotation between two intersecting lines or surfaces.
A full circle is divided into 360 degrees, meaning a straight line forms an angle of 180 degrees and a right angle is 90 degrees.

In the context of complementary angles, it’s critical to remember that two angles are complementary if their measures add up to exactly 90 degrees.

• This unique characteristic of complementary angles is often harnessed to solve problems involving these angles.
• For instance, if one angle is known, the measure of its complement can be easily calculated by subtracting the known angle from 90 degrees.
• This simple calculation underscores the significance of understanding the concept of angle measurement.

Recognizing and applying these measurements in various contexts simplifies problem-solving and analysis of geometrical transformations.

Problem-Solving Strategy

Successful problem-solving often involves a structured strategy, which can significantly enhance clarity and efficiency in reaching a solution. The five-step problem-solving strategy mentioned is an excellent approach to tackle mathematical challenges:
First, understand the concept clearly and identify what you are asked to find.

In this particular exercise, recognizing the relationship between the angle and its complement is the first crucial step.

• Secondly, translating the problem's verbal description into an equation or a mathematical expression aligns with the formulation step.
• Next, solve for the unknown by manipulating the equation to find the answer.
• This systematic approach minimizes confusion and ensures every aspect of the problem is addressed.
• Lastly, verify your solution to ensure its accuracy.

Adopting such a strategic method not only aids in solving the current problem but also builds valuable skills for solving future, more complex problems.

Algebraic Equations

Algebraic equations are mathematical expressions that use variables, constants, and mathematical operations to represent a statement of equality between two quantities.
In this exercise, the algebraic equation is formulated based on the description of the angle in question.

The equation, \(x = (90 - x) + 60\), efficiently expresses the condition that the angle is 60 degrees more than its complement.

• To solve the equation, algebraic manipulations are performed, such as simplifying the equation and isolating the variable to find its value.
• Firstly, simplification involves combining like terms and reformatting the equation for easier computation.
• Solving the equation step-by-step ensures clear understanding of each operation used.
• This not only aids in acquiring the correct answer but also reinforces algebraic skills essential for solving various mathematical problems.

Proper understanding and manipulation of algebraic equations are integral to effectively handling mathematical scenarios, especially in geometry.