Problem 61 - Algebra A Combined Function

Question

A geodesic dome, based on the design by Buckminster Fuller, is composed of two different types of triangular panels. One of these is an isosceles triangle. In one geodesic dome, the measure of the third angle is \(76.5^{\circ}\) more than the measure of either of the two equal angles. Find the measure of the three angles. (Source: Buckminster Fuller Institute)

Step-by-Step Solution

Answer

The angles are 34.5°, 34.5°, and 111°.

Step 1: Define the Variables

Let the measure of the two equal angles be \( x \). Since the problem states that the third angle is 76.5 degrees more than the two equal angles, we will have the third angle as \( x + 76.5^{\circ} \).

Step 2: Use the Triangle Angle Sum Property

The property states that the sum of the angles in a triangle is always equal to 180 degrees. Therefore, we can write the equation as: \( x + x + (x + 76.5) = 180 \), which simplifies to \( 3x + 76.5 = 180 \).

Step 3: Solve for \( x \)

To solve for \( x \), first isolate \( 3x \) by subtracting 76.5 from both sides: \( 3x = 180 - 76.5 \). This simplifies to \( 3x = 103.5 \). Next, divide both sides by 3: \( x = \frac{103.5}{3} = 34.5 \).

Step 4: Find the Third Angle

Since the third angle is \( x + 76.5 \), substitute \( x = 34.5 \) to find the third angle: \( 34.5 + 76.5 = 111 \).

Step 5: Conclusion

The measures of the three angles in the isosceles triangle are \( 34.5^{\circ} \), \( 34.5^{\circ} \), and \( 111^{\circ} \).

Key Concepts

Isosceles Triangle

An isosceles triangle is a special kind of triangle where two of its sides are equal in length. This unique property also means that two of its angles will be equal as well. In the isosceles triangle from our exercise, these equal angles are the foundation for solving the problem of finding all angle measures.
An isosceles triangle has:

• Two equal sides and two equal angles.
• The equal angles are positioned opposite the equal sides.
• The third angle is different unless it forms an equilateral triangle, where all angles and sides are equal.

Understanding the definition of an isosceles triangle is vital for determining the angle measures when other conditions, like in this problem, are provided.

Triangle Angle Sum Property

The triangle angle sum property is a fundamental concept in geometry that states the sum of all internal angles in a triangle always equals 180 degrees. This property is crucial for solving problems involving unknown angle measures, such as the exercise with the isosceles triangle.
Here's how it works:

• For any triangle, add the three internal angles together: their sum will always be 180 degrees.
• This property helps solve for unknown angles when some angles are known.
• It plays a key role in deriving equations needed to find unknown angles in geometric problems.

In our problem, this property allowed us to set up the equation: \[ x + x + (x + 76.5) = 180 \] , simplifying it to solve for the angles of the isosceles triangle.

Angle Measures

Angle measures refer to the specific sizes of each angle within a triangle, typically expressed in degrees. In the case of the isosceles triangle discussed, determining these measures was achieved through a mix of known properties and mathematical calculations.
Steps for finding the angle measures in our scenario included:

• Defining the equal angles as \( x \). Since one angle was given as more than the others, the third angle was \( x + 76.5^{\circ} \).
• Utilizing the triangle angle sum property to form an equation involving these angles.
• Solving this equation to find the specific angle measures.

Through calculation, the measured angles of the triangle were determined to be \( 34.5^{\circ} \), \( 34.5^{\circ} \), and \( 111^{\circ} \). These measures illustrate how known factors can be used to solve for unknowns in geometry.

Problem Solving Steps

Solving geometry problems, like those involving an isosceles triangle, requires a structured approach. Gathering all known information and applying logical steps can lead to the solution.
Here's a breakdown of effective problem-solving steps:

• Define Variables: Assign symbols to the unknowns, such as letting \( x \) represent the equal angles in a triangle.
• Use Known Properties: Apply properties such as the triangle angle sum property to create equations.
• Solve Equations: Manipulate equations to solve for the unknowns. This could involve algebraic techniques like isolating variables.
• Verify Solutions: Check that the solutions make sense within the context of the problem and adhere to all given conditions.

For the isosceles triangle exercise, adhering to these steps enabled the successful determination of the triangle's angle measures. This systematic method is invaluable for tackling a wide range of geometry problems.