Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the relationship among the three angles of any triangle to solve. One angle of a triangle is three times as large as another The measure of the third angle is \(40^{\circ}\) more than that of the smallest angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The measures of the triangle's angles are \(28^{\circ}\), \(68^{\circ}\), and \(84^{\circ}\).

Step 1: Set up the equations

Let's define the angles. Call the smallest angle \(x\), then the other angles would be \(3x\) (since one angle is three times as large as the smallest) and \(x+40\) (because the third angle is 40 degrees more than the smallest). We know that the sum of all angles in a triangle is \(180^{\circ}\), so we can write an equation like this: \(x + 3x + x + 40 = 180\)

Step 2: Solve the equation

Now, we will simplify and solve this equation. By adding the similar terms, the equation becomes \(5x + 40 = 180\). Subtracting 40 from both sides to isolate \(5x\), the equation transforms into \(5x = 140\). Finally, we divide both sides of this equation by 5 to find \(x = 28\)

Step 3: Find all angles

Substitute \(x = 28\) back into the expressions we used to define the angles. So, the smallest angle is \(x = 28^{\circ}\), the largest is \(3x = 3 * 28 = 84^{\circ}\), and the medium one is \(x + 40 = 28 + 40 = 68^{\circ}\).

Key Concepts

Algebraic Equations

Understanding algebraic equations is fundamental when it comes to solving a variety of mathematical problems, including those pertaining to triangle angle problem solving. An algebraic equation is a statement of equality between two mathematical expressions. Constructing and solving equations allows us to find unknown values that make the equation true.

In the context of our triangle problem, we establish a relationship between the angles using variables. Setting the smallest angle as 'x' creates a solid foundation to build upon. To solve an algebraic equation, we often combine like terms, isolate the variable, and then perform basic arithmetic operations. In this case, we combined terms to turn the equation into a manageable form, \(5x + 40 = 180\), and then used subtraction and division to isolate x and find that \(x = 28\).

Following these steps systematically is key to finding a clear path to the solution. For students, remembering to define variables to represent unknown components and utilizing arithmetic operations correctly can greatly simplify these problems.

Triangle Sum Theorem

The triangle sum theorem is a critical concept in geometry that states the sum of the interior angles of any triangle equals \(180^\circ\). This theorem is the backbone of many problems involving triangles, as it applies to every single triangle, regardless of its shape or size.

Utilizing this theorem, we can set up an algebraic equation based on the relationships given between the angles. For instance, knowing that one angle of our triangle problem is three times larger than another and that the third angle is \(40^\circ\) more than the smallest angle, we can express all angles in terms of the smallest angle 'x' and then use the sum of \(180^\circ\).

It's imperative for students to remember this theorem and understand that it applies universally to triangles. In solving problems, the triangle sum theorem is often combined with other properties and theorems to determine unknown angles or sides.

Angle Relationships in Triangles

Within triangles, there are several types of angle relationships that can help solve for unknown measures. These include relationships like complementary and supplementary angles, as well as, in this case, a proportion between angles.

When one angle is described as being a multiple of another, as in our example where an angle is three times as large as the smallest angle, we can use that information to express the angles algebraically. In addition, knowing that the third angle is \(40^\circ\) more than the smallest gives us a clear link between the angles, creating an easily solvable equation when combined with the triangle sum theorem.

Understanding these relationships between angles, along with consistent practice in setting up equations, drastically improves problem-solving skills. Students should identify and apply these relationships to set up correct algebraic equations, leading them to the right answers efficiently.