Question

In a triangle, angle \(B\) is 4 degrees less than twice the measure of angle \(A,\) and angle \(C\) is 11 degrees less than three times the measure of angle \(B .\) Find the measure of each angle.

Short answer

A = 23°, B = 42°, C = 115°.

Step-by-step solution

Define the variables

Let the measure of angle A be denoted as \( A = x \). Since angle B is 4 degrees less than twice the measure of angle A, define angle B as \( B = 2x - 4 \). Since angle C is 11 degrees less than three times the measure of angle B, define angle C as \( C = 3(2x - 4) - 11 \).

Simplify the expression for angle C

Simplify the expression for angle C: \[ C = 3(2x - 4) - 11 \] \[ C = 6x - 12 - 11 \] \[ C = 6x - 23 \].

Use the triangle angle sum property

The sum of the angles in a triangle is always 180 degrees. Therefore, we write the equation: \[ x + (2x - 4) + (6x - 23) = 180 \].

Solve for x

Combine like terms to solve for \( x \): \[ x + 2x - 4 + 6x - 23 = 180 \] \[ 9x - 27 = 180 \] \[ 9x = 207 \] \[ x = 23 \].

Find the measures of angles B and C

Substitute \( x = 23 \) back into the expressions for angles B and C: \[ B = 2(23) - 4 = 46 - 4 = 42 \] \[ C = 6(23) - 23 = 138 - 23 = 115 \].

Write the final answer

Therefore, the measures of the angles are: \[ A = 23^\text{°} \] \[ B = 42^\text{°} \] \[ C = 115^\text{°} \].

Key concepts

angle sum property

In any triangle, the sum of the interior angles is always 180 degrees. This rule is known as the angle sum property. This means that if you know the measures of two angles in a triangle, you can always find the third angle by subtracting the sum of the two known angles from 180. This property is fundamental in geometry and is helpful in solving many problems involving triangles. For example, in this problem, we used the angle sum property to set up our equation by adding the three angles and setting their sum equal to 180 degrees.

algebraic expressions

Algebraic expressions are a way to describe numbers and operations in terms of variables. They allow you to create equations that represent real-world situations. In the given problem, we used algebraic expressions to define the measures of angles B and C based on the measure of angle A. Algebraic expressions involve variables (like x) and constants (like numbers), and can include operations like addition, subtraction, multiplication, and division. They are essential tools in both algebra and geometry.

solving equations

Solving equations is the process of finding the value of variables that make the equation true. In this problem, we set up an equation based on the angle sum property and then solved for the variable x, which represents the measure of angle A. To solve the equation, we combined like terms and isolated the variable on one side of the equation. This gave us the value of x. We then used this value to find the measures of angles B and C by substituting x back into the expressions for B and C.

geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of figures. It includes the study of points, lines, angles, surfaces, and solids. In this problem, the focus was on the properties of a triangle. Geometry helps us understand and solve problems related to these shapes by providing theorems, such as the angle sum property, and tools like algebraic expressions. Understanding geometry is crucial in many fields, including engineering, architecture, and physics.