Problem 81 - Algebra A Combined Function

Question

The sum of the angles of a triangle is \(180^{\circ} .\) If one angle of a triangle measures \(x^{\circ}\) and a second angle measures \((2 x+7)^{\circ},\) express the measure of the third angle in terms of \(x\). Simplify the expression.

Step-by-Step Solution

Answer

The third angle is \(173 - 3x\) degrees.

Step 1: Setup the Equation for Angles of a Triangle

In any triangle, the sum of the interior angles equals \(180^{\circ}\). We are given two angles: \(x^{\circ}\) and \((2x + 7)^{\circ}\). We need to find the third angle, let's call it \(y\). The equation to represent the sum of the angles is: \(x + (2x + 7) + y = 180\).

Step 2: Simplify the Equation

First, combine like terms in the equation. This gives us \(3x + 7 + y = 180\).

Step 3: Solve for the Third Angle

To find the value of \(y\), the third angle, rearrange the equation: \(y = 180 - 3x - 7\). Simplify the expression for \(y\) to get \(y = 173 - 3x\).

Step 4: Simplified Expression

The simplified expression for the measure of the third angle in terms of \(x\) is \(173 - 3x\) degrees.

Key Concepts

Algebraic Expressions

Algebraic expressions are like phrases in a language, but instead of using words, we use numbers, variables, and mathematical symbols to represent relationships and quantities. When you see something like \(x\) or \(2x + 7\), you are looking at an algebraic expression. Here's how it works:

• **Variables:** Letters such as \(x\) represent unknown values that we can solve for.
• **Constants:** Numbers that have a fixed value, like \(7\).
• **Coefficients:** Numbers that multiply the variables, such as \(2\) in \(2x\).
• **Operators:** Symbols like \(+\) and \(-\) are used to connect different parts of an expression.

In the context of the triangle problem, we express angles of triangles using such expressions. \(x\) stands for the first angle, and \(2x + 7\) adds more complexity to the measurement of the second angle. Understanding these expressions helps us manipulate and simplify them to find solutions.

Interior Angles of a Triangle

The interior angles of a triangle are the angles formed inside the triangle. A crucial property of triangles is that the sum of their interior angles always equals \(180^{\circ}\). This principle is foundational in geometry.

• It applies to all triangles, whether they are scalene, isosceles, or equilateral.
• This consistent property aids in solving for unknown angles.

In any given triangle, if you know two of the angles, you can always calculate the third by subtracting the sum of the known angles from \(180^{\circ}\). This property makes solving triangle angle problems systematic and straightforward. For the exercise, knowing \(x^{\circ}\) and \((2x + 7)^{\circ}\) allows us to write an equation to find the unknown third angle.

Solving Equations

Solving equations is a key skill in algebra, allowing us to find unknown values by balancing both sides of an equation. In the context of a triangle's angles, we set up equations that represent the relationships between the angles. Here's how you approach solving an equation:

• **Set Up the Equation:** Identify what you need to find. For example, our equation \(x + (2x + 7) + y = 180\) is set up to find the third angle \(y\).


• **Combine Like Terms:** Group similar terms, simplifying the expression. For instance, combine \(x\) and \(2x\) to get \(3x\), leading to \(3x + 7 + y = 180\).


• **Isolate the Variable:** Rearrange the expression so you get the unknown on one side. Here, subtract \(3x + 7\) from \(180\) to isolate \(y\).


• **Simplify:** Solve the equation to express \(y = 173 - 3x\), giving you a clear formula to find the third angle for any value of \(x\).

Mastering this process ensures you can tackle complex problems by breaking them into manageable steps.