Problem 35 - Introductory Algebra for College Students

Question

One angle of a triangle is twice as large as another. The measure of the third angle is \(20^{\circ}\) more than that of the smallest angle. Find the measure of cach angle.

Step-by-Step Solution

Answer

The smallest angle measures \(40^{\circ}\), the largest angle measures \(80^{\circ}\), and the third angle measures \(60^{\circ}\).

Step 1: Define variables

Let's say the smallest angle is \(x\), the largest angle will be \(2x\) (twice the smallest one), and the third angle is \(20 + x\) (as it is \(20^{\circ}\) more than smallest angle).

Step 2: Write the equation

The sum of the angles in a triangle is \(180^{\circ}\). So, we write the equation as \(x + 2x + (20 + x) = 180\).

Step 3: Simplify and solve

Simplify the equation to \(4x + 20 = 180\). Solve for \(x\), by subtracting 20 from both sides, then dividing by 4, to get \(x = 40\). This is the size of the smallest angle.

Step 4: Determine the other angles

The largest angle is twice the smallest one (\(2 * x = 2 * 40 = 80^{\circ}\)). The third angle is \(20^{\circ}\) more than the smallest one (\( x + 20 = 40 + 20 = 60^{\circ}\)).

Step 5: Check the solution

We add all the found angles. The sum is \(40^{\circ} + 80^{\circ} + 60^{\circ} = 180^{\circ}\), which is the total sum of angles in a triangle, so our solution is correct.

Key Concepts

Understanding Angle Relationships

When solving problems involving triangles, it's essential to understand how angles relate to one another. In this particular problem, we see a direct relationship between the angles where one angle is twice another, and the third is greater by a specific amount from the smallest angle.

Such relationships help in forming equations that can be solved. For example:

• If one angle is expressed in terms of another, it simplifies setting up an equation.
• Identifying patterns, such as when one angle is a multiple of another, can be critical.
• Always keep in mind the triangle angle sum property, which states that all angles together must total 180 degrees.

Recognizing these relationships is a keystone in geometry, as it simplifies the problem-solving process and builds the foundation for future complex applications.

Using Algebraic Equations to Solve Angle Problems

To solve the angle problem in the triangle, algebraic equations come in very handy. Algebra allows us to express unknown values with variables and create equations to find those values.

Consider how the problem is translated into an equation:

• Assign the smallest angle as \(x\).
• Express other angles based on \(x\); for instance, double of the smallest and 20 degrees more than the smallest.
• Construct the equation \(x + 2x + (x + 20) = 180\) and solve for \(x\).
  This equation captures the essence of the relationships and uses the angle sum property.

The beauty of using algebra is in its universality—you can apply it to solve for any number of related angles, regardless of the additional complexities.

The Role of Triangle Properties

Triangle properties are foundational in understanding and solving geometric problems. The most fundamental property is that the sum of the internal angles of any triangle is always 180 degrees.

This rule helps to validate solutions:

• When you calculate each angle, always check if their sum equals 180 degrees.
• If a problem involves complementary or supplementary angles within triangles, these should also adhere to the sum rule.

Furthermore, knowing the specific properties of different types of triangles, such as isosceles or equilateral, can offer additional insights:

• For instance, isosceles triangles have two equal angles.
• In equilateral triangles, all angles are equal to 60 degrees.

Understanding these properties not only aids in solving basic problems but prepares you for more advanced geometry concepts.