Problem 36 - Algebra A Combined Function

Question

The measure of the largest angle of a triangle is \(90^{\circ}\) more than the measure of the smallest angle, and the measure of the remaining angle is \(30^{\circ}\) more than the measure of the smallest angle. Find the measure of each angle.

Step-by-Step Solution

Answer

The angles are \(20^{\circ}, 50^{\circ}, 110^{\circ}\).

Step 1: Define the variables

Let's denote the smallest angle of the triangle as \( x \). The problem states the largest angle is \( 90^{\circ} \) more than the smallest angle, so we can express the largest angle as \( x + 90^{\circ} \). Similarly, the remaining angle is \( 30^{\circ} \) more than the smallest angle, which can be written as \( x + 30^{\circ} \).

Step 2: Use the triangle angle sum property

The sum of the interior angles in a triangle is always \( 180^{\circ} \). Thus, we can write an equation based on this property: \[ x + (x + 90^{\circ}) + (x + 30^{\circ}) = 180^{\circ} \]

Step 3: Simplify the equation

Combine like terms in the equation:\[ 3x + 120^{\circ} = 180^{\circ} \]

Step 4: Solve for \( x \)

Subtract \(120^{\circ} \) from both sides of the equation:\[ 3x = 60^{\circ} \]Then, divide both sides by 3 to solve for \( x \):\[ x = 20^{\circ} \]

Step 5: Find the measures of the other angles

Now that we have \( x = 20^{\circ} \), substitute \( x \) back into the expressions for the other angles:- Largest angle: \( 20^{\circ} + 90^{\circ} = 110^{\circ} \)- Remaining angle: \( 20^{\circ} + 30^{\circ} = 50^{\circ} \)

Key Concepts

Angle Sum Property

In any triangle, the sum of the three interior angles always equals \(180^{\circ}\). This is a fundamental concept in geometry and is known as the triangle angle sum property. Understanding this property is essential because it allows us to find unknown angles when given some information about a triangle's angles. For instance, if you know two angles of a triangle, you can easily find the third angle by subtracting the sum of the known angles from \(180^{\circ}\). This property is not only restricted to numerical problems but also plays a vital role in proving various geometric theories in trigonometry and other advanced math fields. Using this concept effectively requires practice in setting up and solving equations, as seen in many geometry problems involving triangles.

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the value(s) of the variable(s) that make the equation true. In our triangle problem, once we define the variables based on the given conditions, we create an equation using the angle sum property. The equation \[ x + (x + 90^{\circ}) + (x + 30^{\circ}) = 180^{\circ} \] represents the relationship between the angles.

• Identify and combine like terms to simplify the equation.
• Perform basic arithmetic operations such as addition, subtraction, multiplication, and division to isolate the variable.

This method of simplifying and rearranging helps solve the problem step-by-step, making sure every operation follows logically from the previous one. After simplification, you reach the solution \(x = 20^{\circ}\) which gives us meaningful results about the triangle's angles.

Triangle Geometry

Geometry involving triangles is a rich and fascinating field of study that covers various properties and types of triangles. Triangles have three sides and three angles, and their properties can often be used to solve problems about shapes and space. This specific problem explored a few basic concepts of triangle geometry by dealing with an acute angle and calculating other angles using given relationships.

• The smallest angle was denoted as \(x\).
• The largest angle depended on the smallest angle, being \(90^{\circ}\) greater.
• The remaining angle was \(30^{\circ}\) more than the smallest angle.

Understanding how to utilize these relationships and connect them using the angle sum property allows solving these kinds of problems effectively. Triangle geometry isn't only about calculating; it's about using logic and mathematical proof to understand dimension and form.