Problem 26 - Prealgebra

Question

One angle in a triangle measures twice the smallest angle, while the largest angle is six times the smallest angle. Find the measures of all three angles.

Step-by-Step Solution

Answer

The angles are 20°, 40°, and 120°.

Step 1: Assign Variables

Let's denote the smallest angle as \( x \). According to the problem, one angle is twice the smallest angle, so let's call that angle \( 2x \). The largest angle is six times the smallest angle, so let's denote that as \( 6x \).

Step 2: Use Triangle Angle Sum Property

The sum of all angles in a triangle is always \( 180^\circ \). Therefore, you can set up the equation: \( x + 2x + 6x = 180 \).

Step 3: Simplify the Equation

Combine all terms in the equation: \( 1x + 2x + 6x = 9x = 180 \).

Step 4: Solve for \( x \)

Solve the equation \( 9x = 180 \) by dividing both sides by 9. This gives \( x = 20 \).

Step 5: Calculate Each Angle

Now that we know \( x = 20 \), the smallest angle is \( 20^\circ \). The next angle is \( 2x = 2 \times 20 = 40^\circ \). The largest angle is \( 6x = 6 \times 20 = 120^\circ \).

Key Concepts

Angle Sum Property

In every triangle, the sum of all three interior angles always adds up to exactly 180 degrees. This rule is known as the angle sum property of triangles. This is a fundamental property applicable to all types of triangles, whether they are scalene, isosceles, or equilateral.

• Knowing that the angles add up to 180 degrees helps us determine unknown angles when we already have information about some angles.
• In this exercise, the angle sum property serves as a critical tool because it enables us to write an equation involving all the angles of the triangle.

With this property, although you might only know the relationships between the angles, you can calculate the specific measures.

Assigning Variables in Equations

Assigning variables to unknown values is a useful technique for solving problems in geometry and algebra. Here, we need to find the measures of three angles in a triangle where the measures have specific relationships:

• The smallest angle is designated as \( x \) since its measure is unknown.
• Another angle is given by the relationship that it is twice the smallest angle, so this angle is \( 2x \).
• The largest angle in the triangle is six times the smallest angle, assigned the expression \( 6x \).

By representing these angles with variables, it becomes straightforward to write an equation that captures the problem condition, enabling problem-solving through algebra.

Simplifying Equations

Simplifying equations is a step where you combine like terms to reduce complexity and make the equation easier to solve. In our case, once we have assigned variables, the equation becomes: \[ x + 2x + 6x = 180 \]

• Here, \( x \), \( 2x \), and \( 6x \) are like terms since all include the variable \( x \).
• We combine these terms to form \( 9x \), giving us the equation \( 9x = 180 \).

This simplification step is crucial as it condenses the equation and makes it easier to find the solution for \( x \).

Solving Linear Equations

After the equation is simplified, you proceed to solve for the variable. In our situation, we have the linear equation \( 9x = 180 \). Solving this involves a simple arithmetic operation:

• Divide both sides of the equation by 9 to isolate \( x \).
• This results in \( x = 20 \).

Now that we know \( x = 20 \), we can substitute back to find each angle:

• The smallest angle is \( 20^\circ \).
• The next angle is \( 2 \times 20 = 40^\circ \).
• The largest angle is \( 6 \times 20 = 120^\circ \).

Solving linear equations like this helps derive specific numerical answers from an algebraic equation.