Problem 234 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. The angles in a triangle are such that one angle is \(20^{\circ}\) more than the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

Step-by-Step Solution

Answer

The angles are \(32^{\text{°}}\), \(52^{\text{°}}\), and \(96^{\text{°}}\).

Step 1: Identify Variables

Let the smallest angle be denoted as \(x\).

Step 2: Express Other Angles in Terms of the Smallest Angle

According to the problem, one angle is \(20^{\text{°}}\) more than the smallest angle, so the second angle can be written as \(x + 20^{\text{°}}\). The third angle is three times the smallest angle, so it can be written as \(3x\).

Step 3: Use Triangle Angle Sum Property

The sum of the angles in any triangle is \(180^{\text{°}}\). Therefore, the equation will be: \[ x + (x + 20^{\text{°}}) + 3x = 180^{\text{°}} \]

Step 4: Simplify the Equation

Combine like terms to simplify the equation: \[ 5x + 20^{\text{°}} = 180^{\text{°}} \]

Step 5: Solve for the Smallest Angle

Subtract \(20^{\text{°}}\) from both sides to isolate the term with \(x\): \[ 5x = 160^{\text{°}} \]. Then divide by 5: \[ x = 32^{\text{°}} \]

Step 6: Find the Measures of the Other Angles

Now that \(x = 32^{\text{°}}\), substitute back to find the other angles. The second angle is \(32^{\text{°}} + 20^{\text{°}} = 52^{\text{°}}\) and the third angle is \(3 \times 32^{\text{°}} = 96^{\text{°}}\).

Key Concepts

Triangle Angle Sum

In every triangle, the sum of the three interior angles is always equal to 180 degrees. This is a fundamental property. Think of it like a rule that never changes, no matter the size or shape of the triangle.

If you know two of the angles, you can always find the third by subtracting the sum of the known angles from 180 degrees. For example, if two angles are 50 and 60 degrees, the third angle would be 180 - (50 + 60) = 70 degrees.

In our exercise, we used this rule to set up an equation with the three angles. This is the key starting point for solving the problem.

Algebraic Expressions

Understanding and using algebraic expressions is crucial in solving geometry problems. An expression like \(x + 20^{\circ}\) represents an unknown angle that is 20 degrees more than another angle \(x\).

In our exercise, we denoted the smallest angle as \(x\). We created expressions for the other two angles based on the information given. The second angle is \(x + 20^{\circ}\) and the third angle is \(3x\). These expressions allow us to form an equation that we can solve.

Breaking down the problem into manageable pieces with expressions makes it easier to see how each part relates to the others.

Solving Equations

Solving equations involves finding the values of unknown variables that make the equation true. In geometry problems involving triangle angles, these equations usually come from the triangle angle sum property.

For our exercise, we set up the equation \x + (x + 20^{\circ}) + 3x = 180^{\circ}\. This combines all our expressions. Next, we simplified this to \5x + 20^{\circ} = 180^{\circ}\.

To isolate \(x\), we subtracted 20 from both sides: \5x = 160^{\circ}\. Finally, we divided by 5, giving us \x = 32^{\circ}\. Solving these steps accurately is essential for finding the right angle measures.

Geometry Problem-Solving

Geometry problems often require a step-by-step approach to break down the problem. Start by identifying what you know and what you need to find. Then, use properties and rules like the triangle angle sum to create equations.

For our exercise, after finding the smallest angle \(x\), we used our earlier expressions to find the other angles: \(32^{\circ} + 20^{\circ} = 52^{\circ}\) and \3 \times 32^{\circ} = 96^{\circ}\.

Reviewing each step and checking your work can prevent errors. This methodical approach is valuable not just for triangles, but for solving a wide range of geometry problems.