Problem 48 - 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 \(16^{\circ}\) more than triple that of its supplement.

Step-by-Step Solution

Answer

The measure of the angle is \(139^{\circ}\).

Step 1: Identify the Unknowns and the Known

Let the measure of the angle be \(x^{\circ}\). According to the given statement, the angle is \(16^{\circ}\) more than triple the measure of its supplement. The supplement of the angle is \(180 - x^{\circ}\).

Step 2: Formulate the Equation

From the given statement, it is known that the angle \(x\) is \(16^{\circ}\) more than three times the measure of its supplement. This can be expressed as an equation: \(x = 3(180 - x) + 16\).

Step 3: Simplify the Equation

Expanding the right side of the equation gives \(x = 540 - 3x + 16\). This further simplifies to \(x + 3x = 540 + 16\). Combine like terms to find \(4x = 556\).

Step 4: Solve the Equation

To find the value of \(x\), divide both sides of the equation by 4: \(x = 556 \div 4\). This gives \(x = 139\). This means that the angle's measure is \(139^{\circ}\).

Step 5: Confirm the Solution

To confirm the solution, substitute the measure \(x = 139^{\circ}\) into the original equation and check to see if both sides are equal. The left side of equation is \(x = 139\) and the right side after substitution becomes \(3(180 - 139) + 16 = 139\). Hence, the solution is correct.

Key Concepts

Supplementary Angles

In geometry, when you hear about supplementary angles, it means you are dealing with two angles whose measures add up to exactly 180 degrees. When two angles are supplementary, they are essentially "helpers" to reach that 180-degree mark, much like how two complementary angles add up to 90 degrees.
For example, if you have one angle measuring 60 degrees, its supplementary angle would have to be 120 degrees because 60 + 120 = 180.
This concept is critical when solving angle problems because it allows us to find missing angle measures based simply on given information about their relationships to one another.

Equation Solving

Equation solving is like detective work for mathematicians. It involves finding the value of an unknown variable, often represented by a letter such as \(x\). In our problem, we needed to determine the measure of an angle given its relationship to its supplementary angle. We set up an equation based on this relationship:
\[x = 3(180 - x) + 16\]
Here, \(x\) represents the angle we are trying to find. The number 3 indicates the triple of its supplementary, and 16 is added because the angle is 16 degrees more than this amount. Solving equations can often feel like untangling a puzzle, where simplifying and isolating the variable is key to finding the solution.

Problem-Solving Strategy

Having a clear problem-solving strategy can make math less confusing and more approachable. In this exercise, we applied a five-step strategy to figure out the measure of an angle.

• First, identify the unknowns and what information is provided. In our case, the unknown was the angle, \(x\).


• Next, formulate an equation based on the problem's description, which in this scenario was "16 more than triple its supplement."


• Simplify the equation to make solving easier. This involved combining like terms and managing negative signs accurately.


• Solve the equation for the unknown variable by performing operations that simplify to the solution.


• Finally, confirm the solution by substituting back to check if it fulfills the initial condition, ensuring no mistakes were made.

These steps act as a guide, helping you navigate through various math problems efficiently.