Question

The measures of the angles of a triangle are \(x+2,4 x+3\), and \(x+7\). Find the measure of each angle.

Short answer

The angles are 30°, 115°, and 35°.

Step-by-step solution

Understand the Problem

We have a triangle with angles measured in terms of variable \( x \): \( x + 2 \), \( 4x + 3 \), and \( x + 7 \). We need to find the measure of each angle.

Set Up the Equation

The sum of angles in a triangle is always \( 180^\circ \). Form an equation based on the sum of the angles: \((x + 2) + (4x + 3) + (x + 7) = 180\).

Simplify and Solve the Equation

Combine like terms: \( x + 4x + x + 2 + 3 + 7 = 180 \). This simplifies to \( 6x + 12 = 180 \).

Rearrange and Solve for \( x \)

Subtract 12 from both sides: \( 6x = 168 \). Then, divide both sides by 6 to solve for \( x \): \( x = 28 \).

Calculate Each Angle Measure

Substitute \( x = 28 \) back into the expressions for each angle. The first angle is \( x + 2 = 28 + 2 = 30 \). The second angle is \( 4x + 3 = 4(28) + 3 = 112 + 3 = 115 \). The third angle is \( x + 7 = 28 + 7 = 35 \).

Verify the Solution

Add all angle measures to ensure they sum to \( 180^\circ \): \( 30 + 115 + 35 = 180 \). The solution is verified.

Key concepts

Triangle Angle Sum

When studying triangles, one of the fundamental properties to understand is that the sum of all internal angles in a triangle is always 180 degrees. This concept is crucial for solving many geometric problems, including those involving angles expressed with variables.

The Triangle Angle Sum is a vital rule for both simple and complex triangles. By knowing this rule, you can set up equations to find unknown angle measures. For example, if you're given the angles of a triangle as algebraic expressions, you can still determine their values by setting up an equation where the sum equals 180 degrees.

Using this knowledge allows you to tackle a variety of triangular problems easily and ensures your understanding of the relationships between the angles. Remember:

• The sum of the angles: \( A + B + C = 180^\circ \)
• Knowing two angles always allows you to find the third.
• This principle holds true for all triangles, regardless of type.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operational symbols. It can represent a specific value based on the variables involved. In geometry, you often see angles in triangles expressed using algebraic expressions, as this can simplify complex problems.

When dealing with algebraic expressions, you will encounter terms such as variables, coefficients, and constants.

• Variables are symbols, often \( x \) or \( y \), representing unknown numbers.
• Coefficients are numbers that multiply the variables, such as the "4" in \( 4x \).
• Constants are numbers on their own, like "3" or "7" in expressions you might see.

Understanding how to manipulate these expressions—like combining like terms or applying the distributive property—is key to solving problems. In the context of triangles, this might involve setting multiple expressions equal to 180 degrees and solving for variables to find the measure of each angle.

Variable Substitution

Variable substitution is a method used in algebra to solve equations by replacing a variable with a known value. In geometry problems involving triangles, once you solve for the variable, you substitute this value back into the expressions to find the actual measures of the angles.

The process often follows these steps:

• Identify the expression for each triangle angle, e.g., \( x+2 \), \( 4x+3 \), and \( x+7 \).
• Set up an equation that sums the expressions to 180 degrees. This equation helps find the unknown variable.
• Solve the equation to find the value of the variable, e.g., \( x = 28 \).
• Substitute the known variable back into the expressions to get actual angle sizes.

This technique not only simplifies the problem-solving process but also provides a clear path to verifying your solutions, ensuring they satisfy all given conditions, such as the Triangle Angle Sum property.