Exploring the Angles of a Triangle with a Unique Ratio

In the realm of geometry, triangles can possess various interesting characteristics, particularly when their interior angles follow a specific ratio. In this article, we delve into a triangle where the angles are in the ratio 1:3. We'll explore how to find the measure of each angle, providing a detailed analysis for clarity and understanding.

Understanding the Problem

The problem at hand involves determining the measures of the angles in a triangle where the angles are in the ratio 1:3:3:5. We begin by recognizing a critical property of triangles: the sum of the interior angles is always 180 degrees.

Formulating the Equation

Let's denote the measure of the smallest angle as 'k'. Given the ratio 1:3:3:5, the other angles can be expressed as 3k, 3k, and 5k. The sum of these angles equals 180 degrees:

Equation Setup

[ k 3k 3k 5k 180^circ ]

This simplifies to:

Simplifying the Equation

[ 12k 180^circ ]

By dividing both sides by 12, we solve for 'k':

Solving for 'k'

[ k frac{180^circ}{12} 15^circ ]

Calculating the Angles

With the value of 'k' determined, we can now find the measures of each angle:

The Angles

[ text{Smallest angle} k 15^circ ]

[ 3k 3 times 15^circ 45^circ ]

[ 5k 5 times 15^circ 75^circ ]

Therefore, the measures of the angles are 15°, 45°, and 75°, which collectively sum up to 135°. However, there was a slight misinterpretation in the original problem. The angles should sum up to 180°, so let's correct this:

Correcting and Verifying the Solution

Revisiting the equation setup, we have:

Correct Equation

[ k 3k 3k 5k 180^circ ]

This simplifies to:

Final Calculation

[ 9k 180^circ ]

Dividing both sides by 9, we get:

Correct 'k' Value

[ k frac{180^circ}{9} 20^circ ]

Therefore, the measures of the angles are:

Correct Angles

[ text{Smallest angle} 20^circ ]

[ 3k 3 times 20^circ 60^circ ]

[ 5k 5 times 20^circ 100^circ ]

Conclusion

In conclusion, we have explored how to find the measures of the angles in a triangle when they follow a particular ratio. By solving the equation and verifying the sum of the angles, we determined that the angles in this specific triangle are 20°, 60°, and 100°. This understanding can be applied to various geometric problems and helps in reinforcing the fundamental properties of triangles.

Keywords:

Triangle, Angle Ratio, Geometric Properties