Why do the angles in a triangle always add to 180
The correct answer and explanation is:
The angles in a triangle always add up to 180 degrees because of the fundamental properties of Euclidean geometry, which applies to flat surfaces. This result can be explained through several methods, but a common approach is using parallel lines.
Consider any triangle. If you extend a line from one of its sides, making it a straight line, this forms a supplementary angle (which means the sum of the angles on a straight line is always 180 degrees). Now, if you draw a line parallel to this side through the opposite vertex, you create alternate interior angles. These alternate interior angles are equal to the angles of the triangle at the vertex, due to the properties of parallel lines and transversals.
This helps to establish that the sum of the three angles inside the triangle must equal 180 degrees. The reasoning stems from the fact that any angle outside the triangle, formed by the extension of a side, is supplementary to the angle inside the triangle. These exterior angles, together with the interior angles, must always sum to 180 degrees.
This relationship is consistent across all triangles, whether they are scalene, isosceles, or equilateral. The basic principles of Euclidean geometry ensure that this rule holds true universally. Therefore, regardless of the shape or size of a triangle, the sum of the interior angles will always be exactly 180 degrees.
This property is a key element of the geometry of flat surfaces, and it does not apply in non-Euclidean geometry (such as on curved surfaces), where the sum of angles in a triangle can be more or less than 180 degrees.