When Can the Third Angle of a Triangle be 60 Degrees if One Angle is Twice the Size of Another?

The question of whether the third angle of a triangle is always 60 degrees when one angle is twice the size of another is often mis understood. To explore this, we will delve into the geometric properties and the angle sum property of triangles, providing clear explanations and examples.

The Angle Sum Property of Triangles

A fundamental property of triangles is that the sum of their interior angles is always 180 degrees. This is known as the angle sum property. Given two angles in a triangle, the third angle can be determined by subtracting the sum of the given angles from 180 degrees.

Set Up the Problem

Let's denote the smaller angle as ( x ). Consequently, the other angle that is twice the size will be ( 2x ). According to the angle sum property of triangles, the sum of the angles in a triangle is 180 degrees:

[ x 2x text{third angle} 180 ]

This simplifies to:

[ 3x text{third angle} 180 ]

From this equation, we can express the third angle as:

[ text{third angle} 180 - 3x ]

Determine When the Third Angle is 60 Degrees

Now, to find out when the third angle equals 60 degrees, we can set up the equation:

[ 180 - 3x 60 ]

Solving for ( x ):

[ 3x 120 ]

[ x 40 ]

So if the smaller angle ( x ) is 40 degrees, the larger angle ( 2x ) would be 80 degrees, and the third angle would indeed be 60 degrees.

General Case Analysis

However, this is just one specific case. In general, as ( x ) varies, the third angle will change accordingly. Let's illustrate this with some examples:

Example 1

When ( x 30 ) degrees:

[ 2x 60 ] degrees

[ text{third angle} 180 - 90 90 ] degrees

Example 2

When ( x 20 ) degrees:

[ 2x 40 ] degrees

[ text{third angle} 180 - 60 120 ] degrees

General Conclusion

Thus, the third angle can vary and is not always 60 degrees. Let's consider some more examples:

Example: Triangle ABC

1. [ text{Angle A} 100°, text{Angle B} 50°, text{Angle C} 30° ]

2. [ text{Angle A} 50°, text{Angle B} 25°, text{Angle C} 105° ]

3. [ text{Angle A} 70°, text{Angle B} 35°, text{Angle C} 75° ]

4. [ text{Angle A} 64°, text{Angle B} 32°, text{Angle C} 84° ]

In any of the above examples, the angle A is twice the angle B, but the third angle is not 60 degrees.

Special Case

The third angle will be 60 degrees only if the smaller angle ( x ) is 40 degrees, making the larger angle 80 degrees. In all other cases, the third angle will not be 60 degrees.

Conclusion

In summary, the third angle of a triangle is not always 60 degrees when one angle is twice the size of another. The specific condition is that ( x 40 ) degrees, resulting in the larger angle being 80 degrees and the third angle being 60 degrees. The variations in the smaller angle ( x ) lead to different values for the third angle, which can range from 90 to 120 degrees, depending on the value of ( x ).