Question

Algebra The measures of the angles of a triangle are \(x+5\), \(3 x+14\), and \(x+11\). Find the measure of each angle.

Short answer

The angles are 35°, 104°, and 41°.

Step-by-step solution

Understand the Problem

The problem gives us three angles of a triangle in terms of the variable \(x\). The angles are given as \(x+5\), \(3x+14\), and \(x+11\). We need to find the measure of each angle.

Use the Triangle Angle Sum Property

According to the triangle angle sum property, the sum of all the angles in a triangle is always \(180\) degrees. Therefore, we can set up the equation: \((x + 5) + (3x + 14) + (x + 11) = 180\).

Simplify the Equation

Combine like terms in the equation: \(x + 3x + x + 5 + 14 + 11 = 180\). This simplifies to \(5x + 30 = 180\).

Solve for \(x\)

Subtract 30 from both sides of the equation to get \(5x = 150\). Then divide both sides by \(5\) to find \(x = 30\).

Calculate Each Angle

Now that we know \(x = 30\), substitute \(x\) back into the expressions for each angle: 1. The first angle is \(x + 5 = 30 + 5 = 35\) degrees.2. The second angle is \(3x + 14 = 3(30) + 14 = 104\) degrees.3. The third angle is \(x + 11 = 30 + 11 = 41\) degrees.

Verify the Solution

Verify that the sum of the angles is 180 degrees: \(35 + 104 + 41 = 180\). Since the sum is 180, the solution is correct.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with variables, equations, and the relationship between them. When working with algebra problems, we often have expressions containing unknown variables that we need to solve for. In the context of this exercise, we are given expressions for the angles of a triangle in terms of the variable \(x\):

• \(x + 5\)
• \(3x + 14\)
• \(x + 11\)

These expressions involve adding, multiplying, and possibly rearranging terms to isolate and find the value of \(x\). Algebra allows us to handle these expressions systematically and solve for unknowns by utilizing mathematical properties and rules.

Solving Equations

Solving equations is an essential skill in algebra that involves finding the value of unknown variables that make an equation true. In our problem, we used the equation derived from the triangle angle sum property: \((x + 5) + (3x + 14) + (x + 11) = 180\). Solving this equation involves several key steps:

• Combine like terms to simplify the equation: \(x + 3x + x + 5 + 14 + 11 = 180\) becomes \(5x + 30 = 180\).
• Solve for \(x\) by isolating it on one side. Subtract 30 from both sides to get \(5x = 150\).
• Divide both sides by 5 to find \(x = 30\).

These steps require you to perform basic arithmetic operations while maintaining equality. Always remember to check your solution by substituting back into the original equation to ensure that it holds true.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of figures. In particular, triangles are a fundamental part of geometry. A key property of triangles is that the sum of their interior angles is always \(180\) degrees. This is known as the triangle angle sum property.In our exercise, understanding this geometric property was crucial to setting up our equation using the expressions for the angles of the triangle. By knowing that the sum of the angles must be \(180\), we could deduce the equation we needed to solve. Once we found \(x\), we substituted it back into the expressions to determine the exact measures of each of the triangle's angles:

• First angle: \(x + 5 = 35\) degrees
• Second angle: \(3x + 14 = 104\) degrees
• Third angle: \(x + 11 = 41\) degrees

These calculations not only highlight the importance of the angle sum property but also connect the concepts of geometry and algebra in problem-solving.