Solving Triangle Angles with Algebra: A Comprehensive Guide

Triangle angles are a fundamental concept in geometry, and understanding how to solve for unknown angles is crucial for various applications in mathematics and real-world problems. This article provides a detailed guide on solving for the angles of a specific triangle where the conditions are given. We will break down the problem into manageable steps and use algebraic equations to find the solution. Let's get started!

Problem Statement

The largest angle of a triangle is twice the sum of the other two angles, and the smallest angle is one-sixth of the largest angle. What are the angles of the triangle in degrees?

Step-by-Step Solution

Step 1: Define the Angles

Let's denote the angles of the triangle as A, B, and C, where A is the largest angle, B is the smallest angle, and C is the remaining angle.

Step 2: Set up the Given Conditions

The largest angle A is twice the sum of the other two angles:

A  2(B   C)

The smallest angle B is one-fourth of the largest angle:

B  (frac{1}{4}A)

The sum of all angles in a triangle is 180°:

A   B   C  180°

Step 3: Substitute the Second Condition into the First

Substitute B (frac{1}{4}A) into the first condition:

A  2(left(frac{1}{4}A   Cright))

Simplify this equation:

A  (frac{1}{2}A   2C)

Subtract (frac{1}{2}A) from both sides:

(frac{1}{2}A  2C)

Solve for C:

C  (frac{1}{4}A)

Step 4: Substitute C and B into the Angle Sum Equation

We now have:

B (frac{1}{4}A) C (frac{1}{4}A)

Substitute these into the angle sum equation:

A   B   C  180°

Combine like terms:

A   (frac{1}{4}A   (frac{1}{4}A  180°)

Simplify and solve for A:

A   (frac{1}{2}A  180°)

(frac{3}{2}A  180°)

A  (frac{180° times 2}{3}  120°)

Now, find B and C:

B  (frac{1}{4}A  (frac{1}{4} times 120°  30°)

C  (frac{1}{4}A  (frac{1}{4} times 120°  30°)

Conclusion

The angles of the triangle are A 120°, B 30°, and C 30°.

Alternative Methods

Let's explore another method to solve the same problem for varied understanding:

Method 1: Simplified Approach

Let A B and C be the angles of the triangle. A is the largest and C is the smallest.
A  6C
A  B   C OR B  5C
A   B   C  180°
6C   5C   C  180°
12C  180° OR C  180/12  15°
C  15° B  15 times 5  75° A  15 times 6  90°

The angles are A 90°, B 75°, and C 15°.

Conclusion

Understanding how to solve triangle angles using algebra is a valuable skill. Whether you adopt the method shown in this article or another simplified approach, the key is to apply the given conditions and algebraic equations systematically. By following these steps, you can find the angles of the triangle accurately.