Question

Prove that the sum of interior angles of a triangle is 180 degrees.

Answer 1

To prove that the sum of the interior angles of a triangle is 180 degrees, we can use a simple geometric approach involving parallel lines. Here’s a step-by-step proof:

1. **Consider a triangle**: Let triangle \( ABC \) have interior angles \( \angle A \), \( \angle B \), and \( \angle C \).

2. **Extend one side**: Extend the side \( BC \) of the triangle to a point \( D \) beyond \( C \).

3. **Draw a parallel line**: Through point \( A \), draw a line parallel to line \( BC \). Let this line intersect line \( AD \) at some point.

4. **Identify angles**: Since line \( AD \) is a transversal intersecting the two parallel lines \( BC \) and the line through \( A \), we can use the properties of alternate interior angles:
- \( \angle CAB \) (which is \( \angle A \)) is equal to the angle formed at \( A \) by the transversal and the parallel line, let's call it \( \angle EAD \).
- \( \angle ABC \) (which is \( \angle B \)) is equal to the angle \( \angle ABD \) formed at point \( B \) by the transversal \( AD \).

5. **Sum of angles**: Now we observe that the angles on a straight line must sum to 180 degrees:
\[
\angle EAD + \angle ABC + \angle ACB = 180^\circ
\]

6. **Replace angles**: Now substituting the equal angles we identified earlier into this equation gives:
\[
\angle A + \angle B + \angle C = 180^\circ
\]

7. **Conclusion**: Hence, we have proved that the sum of the interior angles of triangle \( ABC \) is indeed 180 degrees.

\(\boxed{180^\circ}\)