The sum of all three interior angles in any triangle is always one hundred eighty degrees. This is a fundamental theorem in Euclidean geometry that applies to all triangles, regardless of their shape or size.
We can prove this theorem using parallel lines. First, draw a line through vertex C parallel to side AB. The angles on this straight line sum to one hundred eighty degrees. Since alternate interior angles are equal, angle A equals the left angle at C, and angle B equals the right angle at C. Therefore, the sum of all three triangle angles equals one hundred eighty degrees.
The one hundred eighty degree rule applies to all types of triangles. In an acute triangle where all angles are less than ninety degrees, they still sum to one hundred eighty degrees. In a right triangle with one ninety degree angle, the sum is still one hundred eighty degrees. Even in an obtuse triangle with one angle greater than ninety degrees, the total is always one hundred eighty degrees. This is a universal property of all triangles.
Let's see an interactive demonstration. As we change the shape of the triangle by adjusting one angle, the other angles automatically adjust to maintain the one hundred eighty degree sum. Watch how angle A changes from sixty degrees to one hundred twenty degrees, while angles B and C adjust accordingly. No matter how we deform the triangle, the total always remains exactly one hundred eighty degrees.
To summarize what we have learned: The sum of interior angles in any triangle is always one hundred eighty degrees. This universal rule applies to all triangle types, whether acute, right, or obtuse. We can prove this theorem using parallel line properties and alternate interior angles. This fundamental principle in Euclidean geometry helps us solve many geometric problems and is essential knowledge for understanding triangular relationships.