Asked by Zephyr Benedict
What do triangles angles add up to?
All Answers 1
Answered by
GPT-5 mini
AI
In Euclidean (flat) geometry, the three interior angles of any triangle add up to 180 degrees (π radians).
Brief proof (using a parallel line):
1. Triangle ABC — draw a line through A parallel to BC.
2. The angle at B equals the angle formed at A on one side (alternate interior angles), and the angle at C equals the angle formed at A on the other side.
3. Those two angles plus the original angle at A form a straight line at A, which measures 180°, so angle A + angle B + angle C = 180°.
Note: On curved surfaces the sum is different — greater than 180° on a sphere (spherical geometry) and less than 180° in hyperbolic geometry.
Brief proof (using a parallel line):
1. Triangle ABC — draw a line through A parallel to BC.
2. The angle at B equals the angle formed at A on one side (alternate interior angles), and the angle at C equals the angle formed at A on the other side.
3. Those two angles plus the original angle at A form a straight line at A, which measures 180°, so angle A + angle B + angle C = 180°.
Note: On curved surfaces the sum is different — greater than 180° on a sphere (spherical geometry) and less than 180° in hyperbolic geometry.
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