Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Find the measure of each angle. Complementary angles with measures \(9 z+6\) degrees and \(3 z\) degrees.

Step-by-Step Solution

Answer

The angles measure 69 degrees and 21 degrees.

Step 1: Understanding Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. We are given two angles that are complementary: one measures \(9z + 6\) degrees, and the other measures \(3z\) degrees.

Step 2: Set Up the Equation

Since these angles are complementary, their sum is 90 degrees. Therefore, we can set up the equation: \((9z + 6) + 3z = 90\).

Step 3: Solve for z

Combine like terms in the equation: \(12z + 6 = 90\). Subtract 6 from both sides to get \(12z = 84\). Then, divide both sides by 12 to solve for \(z\): \(z = 7\).

Step 4: Calculate the Angles

Substitute \(z = 7\) back into the expressions for each angle. For the angle \(9z + 6\), calculate \(9(7) + 6 = 63 + 6 = 69\) degrees. For the angle \(3z\), calculate \(3(7) = 21\) degrees.

Step 5: Verify the Solution

Add the measures of both angles, \(69 + 21 = 90\) degrees, confirming that they are complementary as their sum is 90 degrees.

Key Concepts

Angle Measurement

When we talk about angle measurement, we are simply referring to how we determine the size of an angle in degrees. Complementary angles are a special pair of angles whose measures add up to 90 degrees. This means that if we know one angle, we can easily find its complement by subtracting the known angle from 90. For example, if an angle measures 30 degrees, its complementary angle would be 60 degrees because 30 plus 60 equals 90.
In exercises, the measurements are often given as algebraic expressions, so understanding how to work with these expressions is crucial. When dealing with algebraic expressions, it's essential to ensure that you correctly interpret and perform operations to find accurate measurements of each angle.
To better understand, visualize the angles on a piece of paper. Draw two angles next to each other, forming a right angle. This visual representation helps in comprehending how two measurements can add up to form a right angle.

Equation Solving

Solving equations is like solving a puzzle where you need to find the unknown piece. In the context of complementary angles, you are given expressions for each angle and need to solve for a variable, often represented as "z" in this case.
Here, the expressions for the angles are given as:

• First angle:(9z + 6) degrees
• Second angle:3z degrees

These expressions are set up in an equation because their sum equals 90 degrees. Once you have the equation, like \((9z + 6) + 3z = 90\) , combine like terms to simplify.
This process involves rearranging and simplifying expressions through basic operations such as addition, subtraction, multiplication, or division. Solving the equation helps to find the value of "z," which can be used to determine the exact measurements of the angles.
Using equations in geometry allows flexibility and can solve a variety of problems efficiently by applying algebraic concepts.

Algebraic Expressions

Algebraic expressions are a way to represent numbers and operations using variables. In this exercise, expressions like \(9z + 6\) and \(3z\) are used to represent the degrees of angles. The letter "z" is a variable that stands for a number we need to find.
To work with algebraic expressions, it's important to understand terms:

• \(9z\) - Here, 9 is the coefficient, and "z" is the variable.
• \(+6\) - This is a constant term added to the variable term in the expression.
• \(3z\) - Similarly, 3 is the coefficient of the variable "z".

These expressions need to be manipulated and simplified using arithmetic operations to solve the problem.
By substituting the value of "z" back into these expressions, we can find precise measurements for each angle. Understanding how to work with expressions like these forms the backbone of many problems in algebra and geometry alike, allowing for flexible and manageable problem-solving.