Problem 80 - Elementary Algebra

Question

In an isosceles triangle, the measures of two of the angles are equal. The third angle of an isosceles triangle measures \(42^{\circ}\). The total measure of the angles in the triangle are \(180^{\circ}\). Find the measure of each of the equal angles.

Step-by-Step Solution

Answer

Each of the equal angles measures 69 degrees.

Step 1 - Understand the problem

In an isosceles triangle, two of the angles are equal and the sum of all three angles is 180 degrees. We need to find the measure of each of the equal angles, given that the third angle measures 42 degrees.

Step 2 - Express the sum of angles

The sum of the angles in a triangle is always 180 degrees. Let the measure of each of the two equal angles be denoted by \( x \). Since the third angle is given as 42 degrees, we can write the equation: \[ x + x + 42 = 180 \] which simplifies to: \[ 2x + 42 = 180 \]

Step 3 - Solve for x

Subtract 42 from both sides of the equation to isolate the terms with \( x \): \[ 2x + 42 - 42 = 180 - 42 \] which simplifies to: \[ 2x = 138 \] Now, divide both sides by 2 to find \( x \): \[ x = \frac{138}{2} = 69 \]

Step 4 - Conclusion

Therefore, each of the equal angles in the isosceles triangle measures 69 degrees.

Key Concepts

triangle angle sum

Every triangle, regardless of its type, has a unique property: the sum of its interior angles always adds up to 180 degrees. This rule is fundamental in geometry. It’s like a universal law for triangles.

When working with triangles, this property helps us understand their structure and solve for unknown angles. For example, if we know the measures of two angles in a triangle, we can always find the third angle by subtracting the sum of the known angles from 180 degrees.

In the given problem, the sum of the angles is expressed mathematically as: \[ x + x + 42 = 180 \] This equation helped us gather information about the equal angles and ultimately find their measure.

isosceles triangle properties

An isosceles triangle is special because it has two sides of equal length, and two angles of equal measure. So, if you know one of the angles, you can figure out the other two.

In our problem, we are told that the third angle is 42 degrees. Knowing this, and remembering that the sum of all angles in a triangle is 180 degrees, we deduced that the other two angles must be equal. This is the property that we leverage to find the unknown angle measures.

The equation we write based on this property is: \[ x + x + 42 = 180 \] This sets up the foundation for the algebra we need to solve the problem.

basic algebra steps

Solving the problem involves basic algebra steps. First, we set up our equation: \[ x + x + 42 = 180 = \] Then, we combine like terms: \[ 2x + 42 = 180 \] To isolate the variable \[ 2x \], we subtract 42 from both sides: \[ 2x = 138 \] Finally, we divide both sides by 2: \[ x = 69 \] By following these steps, we find that each of the equal angles in the isosceles triangle measures 69 degrees.

These simple algebraic manipulations allow us to solve for the unknown angles efficiently. Mastering these basics can help you solve a variety of problems in mathematics.