Problem 82 - Algebra A Combined Function

Question

A quadrilateral is a four-sided figure (like the one shown in the figure) whose angle sum is \(360^{\circ} .\) If one angle measures \(x^{\circ},\) a second angle measures \(3 x^{\circ},\) and a third angle measures \(5 x^{\circ},\) express the measure of the fourth angle in terms of \(x\). Simplify the expression.

Step-by-Step Solution

Answer

The fourth angle is \(360^{\circ} - 9x\).

Step 1: Understand the Angles Given

In the problem, we have three angles of a quadrilateral. The angles are described as being \(x^{\circ}\), \(3x^{\circ}\), and \(5x^{\circ}\). We need to find an expression for the fourth angle.

Step 2: Write the Equation for Angle Sum

The sum of angles in a quadrilateral is always \(360^{\circ}\). Therefore, we can write the equation as:\[x + 3x + 5x + \text{Fourth Angle} = 360^{\circ}\]

Step 3: Simplify the Known Angles

First, combine the known angles:\[x + 3x + 5x = 9x\]This simplifies the angle portion to \(9x\).

Step 4: Solve for the Fourth Angle

Subtract the sum of the known angles from \(360^{\circ}\) to find the expression for the fourth angle:\[\text{Fourth Angle} = 360^{\circ} - 9x\]

Step 5: Simplify the Expression

Ensure the expression is as simple as possible. The simplified expression for the fourth angle is:\[360^{\circ} - 9x\]

Key Concepts

Understanding Angle Sum in Quadrilaterals

The concept of angle sum is fundamental in understanding various shapes and figures in geometry. A quadrilateral, which is a four-sided polygon, has a unique property regarding the sum of its interior angles. Whenever dealing with any quadrilateral, whether it's regular like a square or irregular, the sum of its interior angles will always be

• \(360^{\circ}\).

This knowledge allows us to solve problems related to finding unknown angles when given some angle measures. In the exercise, knowing that the sum of the angles equals \(360^{\circ}\) is crucial because it enables us to create an equation that includes all the angles of the quadrilateral.
This equation plays a pivotal role in determining unknown angle measures through a process called expression simplification, which we will discuss next.

Simplification of Expressions

Simplifying expressions is an essential skill in algebra that involves combining like terms to make equations clearer and easier to work with. When given multiple terms, especially those involving variables like in this problem, the first step is to collect and simplify these terms.

In the given problem, you have three angles indicated by

• \(x^{\circ}\),
• \(3x^{\circ}\),
• \(5x^{\circ}\).

As they all share the variable \(x\), they can be added directly:

• \(x + 3x + 5x = 9x\).

Simplifying terms helps to reduce complexity, making it easier to isolate unknown values, which you will do in solving algebraic equations.

Solving Algebraic Equations for Unknown Angles

Algebraic equations allow us to solve for unknowns given known constraints. In our problem, after simplifying the expression of the known angle measures to

• \(9x\),

we can include it in the equation for the angle sum of the quadrilateral,

• \(x + 3x + 5x + \text{Fourth Angle} = 360^{\circ}\).

We then recognize this equation can be rewritten, by substituting the simplified sum of known angles:

• \(9x + \text{Fourth Angle} = 360^{\circ}\).

By rearranging and solving for the Fourth Angle, we subtract \(9x\) from \(360^{\circ}\):

• \(\text{Fourth Angle} = 360^{\circ} - 9x\).

Solving such equations requires a clear understanding of moving terms around the equation, always maintaining balance by performing the same operation on both sides.
This step completes the circle back to the expression simplification, ensuring that every angle within the quadrilateral is accounted for, thereby solidifying your approach to solving real-world problems involving algebra and geometry.