Final Answer:
The sum of the angles in a triangle is equal to a straight angle, which measures 180 degrees.
Step-by-step explanation:
In Euclidean geometry, the sum of the angles in any triangle always adds up to 180 degrees. This fundamental principle can be demonstrated through various methods, one of which is the parallel line angle theorem. When a transversal intersects two parallel lines, alternate interior angles are congruent. Applying this theorem, you can extend one side of a triangle to create a parallel line, forming alternate interior angles with the opposite side of the triangle. The resulting angles form a straight line, confirming that the sum of the triangle's angles equals 180 degrees.
Another approach involves dividing a triangle into two right triangles by drawing an altitude from one vertex to the opposite side. Each right triangle has a right angle, and the sum of the other two angles is equal to 90 degrees. Therefore, the sum of the angles in the original triangle is twice 90 degrees, which again equals 180 degrees.
These geometric proofs emphasize the consistency and universality of the triangle angle sum theorem. Whether through parallel lines or right triangles, the result remains the same: the sum of the angles in a triangle is always equivalent to the measure of a straight angle.