Problem 27 - Intermediate Algebra

Question

The sum of the measures of the angles of a triangle is \(180^{\circ}\). The largest angle is twice the smallest angle. The sum of the smallest and the largest angle is twice the other angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The angles are 40°, 60°, and 80°.

Step 1: Define the Variables

Let the smallest angle be \(x\). Then, according to the problem, the largest angle will be \(2x\). Let the other angle be \(y\).

Step 2: Establish the First Equation

Using the fact that the sum of angles in a triangle is \(180^{\circ}\), we can write: \[ x + y + 2x = 180. \] Simplifying gives: \[ 3x + y = 180. \]

Step 3: Establish the Second Equation

According to the problem, the sum of the smallest angle and the largest angle is twice the other angle. This gives us: \[ x + 2x = 2y. \] Simplifying gives: \[ 3x = 2y. \]

Step 4: Solve for y in terms of x

From the second equation \(3x = 2y\), solve for \(y\): \[ y = \frac{3x}{2}. \]

Step 5: Substitute y into the First Equation

Substitute \(y = \frac{3x}{2}\) into the first equation \(3x + y = 180\): \[ 3x + \frac{3x}{2} = 180. \]

Step 6: Simplify the Equation

Multiply everything by 2 to eliminate the fraction: \[ 2(3x) + 3x = 360, \] which simplifies to: \[ 6x + 3x = 360. \] So, \[ 9x = 360. \]

Step 7: Solve for x

Divide both sides by 9 to find \(x\): \[ x = \frac{360}{9} = 40. \]

Step 8: Find the Other Angles

Using \(x = 40\), find the largest angle: \[ 2x = 2(40) = 80. \] And find the other angle \(y\) using \(y = \frac{3x}{2}\): \[ y = \frac{3(40)}{2} = 60. \]

Step 9: Confirm the Solution

Check if the angles add up to \(180^{\circ}\): \[ 40 + 60 + 80 = 180. \] Since they do, the angle measures are correct.

Key Concepts

Algebraic Equations

Algebraic equations are a powerful tool for solving mathematical problems, especially those that involve relationships between different quantities. In this exercise, algebraic equations help us find the measures of the angles in a triangle. First, we define the variables: the smallest angle as \( x \), the largest angle as \( 2x \), and the other angle as \( y \). Using algebra, we establish two equations based on the given conditions:

• The sum of the angles in a triangle is always \( 180^\circ \). So, \( x + y + 2x = 180 \).


• The sum of the smallest and largest angles is twice the other angle, so \( x + 2x = 2y \).

These equations provide us with a system of linear equations we can solve to find the unknown angles. By manipulating these equations, we simplify and solve them step by step. With algebraic skills, we are able to express one variable in terms of another, substitute it back into the equations, and ultimately find numerical solutions.

Angle Relationships

Understanding angle relationships is crucial in geometric problem solving. In this exercise, three angle relationships are essential:

• The sum of angles in any triangle is always \( 180^\circ \). This foundational property provides the basis for setting up our equations.


• The relationship between the largest and smallest angles states that the largest angle is twice the smallest. Hence, if the smallest angle is \( x \), the largest is \( 2x \).


• The sum of the smallest and largest angles being twice the measure of the other angle provides additional information, leading to the equation \( 3x = 2y \).

Using these relationships, we can connect the angles in logical and algebraic ways, allowing us to solve for the actual measurements.

Problem Solving Steps

Solving complex problems involves a systematic approach. This exercise demonstrates the step-by-step method to solve for the angles in the triangle efficiently.

• **Step 1:** Define the variables. Identify what needs to be solved and express unknown quantities using variables, such as \( x \) and \( y \).


• **Step 2:** Set up equations. Use given conditions to write algebraic equations. For instance, use the angle sum property and relationships between angles.


• **Step 3:** Solve the equations. Use substitution or elimination methods. From our setup, solve for one variable (e.g., express \( y \) in terms of \( x \)).


• **Step 4:** Back-substitute to find other variables. Find numerical values by substituting known values back into equations. For instance, once \( x \) is found, calculate \( y \).


• **Step 5:** Verification. Check that all conditions are met and add the angles to verify they sum to \( 180^\circ \).

Following these steps ensures a logical and error-free journey to the solution. It's a clear path that can be applied to similar geometric problems, making mathematics both understandable and manageable.