Problem 48 - Elementary Algebra

Question

One angle of a triangle has a measure of \(45^{\circ}\), and the measures of the other two angles are in the ratio of 2 to 1 . Find the measures of the other two angles.

Step-by-Step Solution

Answer

The other two angles are 45° and 90°.

Step 1: Understand the Problem

We know that one angle of the triangle is \(45^{\circ}\), and the other two angles are in the ratio of 2 to 1. We need to find the measures of these two angles.

Step 2: Use the Triangle Angle Sum Theorem

The sum of the angles in a triangle is always \(180^{\circ}\). Therefore, if one angle is \(45^{\circ}\), the sum of the other two angles will be \(180^{\circ} - 45^{\circ} = 135^{\circ}\).

Step 3: Set Up the Equation Using the Ratio

Let \(x\) be the measure of the smaller angle. Then the other angle, being twice the smaller one, will be \(2x\). According to the angle sum we found in Step 2, we can write the equation: \(x + 2x = 135^{\circ}\).

Step 4: Solve the Equation

Combine like terms in the equation: \(3x = 135^{\circ}\). Divide both sides of the equation by 3 to find \(x\): \(x = \frac{135}{3} = 45^{\circ}\).

Step 5: Find the Measures of the Angles

Since \(x = 45^{\circ}\), the measure of the smaller angle is \(45^{\circ}\). The other angle is twice this, so it is \(2 \times 45^{\circ} = 90^{\circ}\). Thus, the two angles are \(45^{\circ}\) and \(90^{\circ}\).

Key Concepts

Angle Ratio

Understanding angle ratios is crucial when solving problems involving triangles. In our problem, the other two angles of a triangle that is not given as a direct measurement are expressed in an angle ratio of 2 to 1. This means that if one angle is represented by a variable, say \(x\), then the other angle will be twice the size of \(x\), which is represented as \(2x\). This concept helps us compare the sizes of angles relative to each other.
Using angle ratios allows you to translate a relational statement into a mathematical equation. This approach can be seen with common problems such as budgeting money or mixing ingredients but in our case, it is applied to angles. As simple as it sounds, writing angles in terms of a common variable enables us to deal with them using algebraic techniques.

Equation Solving

Equation solving here involves forming and solving a mathematical statement to find the value of unknown variables. We begin by setting up the equation based on the total sum of angles in a triangle and the given angle ratio.
For the problem, using the Triangle Angle Sum Theorem, which states that sum of all interior angles in a triangle is \(180^{\circ}\), and the existing angle ratio, the equation was formed. The equation is setup as: \(x + 2x = 135^{\circ}\), where \(x\) is the measure of the smaller angle.
By combining like terms, we simplify the equation to \(3x = 135^{\circ}\). Solving this, we divide both sides by 3 to isolate \(x\), resulting in \(x = 45^{\circ}\).
This methodical approach in solving equations is part of a broader mathematical discipline, and mastering these techniques enables learners to handle more complicated problems effectively.

Triangles in Algebra

Triangles present a great opportunity to explore algebraic applications in geometry. When working with triangles, often, we use algebra to find unknown angle measures or lengths.
For instance, in the given exercise, the application of algebra was crucial to find the unknown angles using a ratio.
By integrating geometric principles, like the angle sum theorem, and algebraic techniques, learners can derive values that aren't immediately given.

• Geometric Principle: The Triangle Angle Sum Theorem strictly states that the sum of interior angles equals \(180^{\circ}\).
• Algebraic Technique: Using variables to represent unknowns and forming solvable equations.

Combining these methods enhances problem-solving skills and underscores the interconnectedness of different mathematical areas. In essence, triangles in algebra help bridge foundational geometry and algebra concepts, offering a comprehensive path to solving such problems.