Question

Explain how the parallel postulate can be applied to prove the triangle angle sum theorem.

a. By demonstrating that in a non-Euclidean geometry, the triangle angle sum theorem does not hold.
b. By showing that in Euclidean geometry, parallel lines intersect at a single point.
c. By illustrating that the sum of angles in a triangle is always 180 degrees in Euclidean geometry.
d. By providing a counterexample that disproves the triangle angle sum theorem.

Answer 1

The parallel postulate can be used to prove the triangle angle sum theorem by illustrating that the sum of angles in a triangle is always 180 degrees in Euclidean geometry.

Step-by-step explanation:

The parallel postulate can be applied to prove the triangle angle sum theorem by illustrating that the sum of angles in a triangle is always 180 degrees in Euclidean geometry (option c). In Euclidean geometry, the parallel postulate states that parallel lines never intersect. Using this postulate, we can show that the angles in a triangle add up to 180 degrees.

1. Start with a triangle.
2. Draw a line parallel to one side of the triangle passing through the opposite vertex.
3. Using the parallel postulate, we know that the line will not intersect the other two sides.
4. This creates a triangle between the original triangle and the line.
5. Since the line is parallel to one side, the corresponding angles are congruent.
6. So, the sum of the angles in the new triangle is the same as the sum of the angles in the original triangle.
7. Since the new triangle is a line and has a sum of 180 degrees, it follows that the sum of the angles in the original triangle is also 180 degrees.