This theorem is of ch- lines and angles and is given in ncert ......just check it .It is quite easy theorem.

Proof that the sum of three angles of a triangle is two right angles

To prove that the sum of three angles of a triangle is equal to two right angles (180 degrees), we can use Euclid's fifth postulate which states that if a straight line intersects two other straight lines, and the sum of the interior angles on one side is less than two right angles, then the two lines will eventually meet on that side.

Proof:

1. Consider an arbitrary triangle ABC:

Let's assume we have a triangle ABC, where A, B, and C are the three vertices of the triangle, and a, b, and c are the three sides opposite to each vertex, respectively.

2. Draw a line parallel to side BC from vertex A:

Draw a line DE parallel to side BC from vertex A, such that it intersects the extension of side AB at point E.

3. Measure the angles:

Measure the angles of the triangle ABC. Let's denote angle BAC as α, angle ABC as β, and angle BCA as γ.

4. Measure the angles created by the parallel line:

Since line DE is parallel to line BC, the alternate interior angles are congruent. Therefore, angle CED is congruent to angle BCA (γ), and angle DEA is congruent to angle ABC (β).

5. Observe the sum of angles:

Now, we have a triangle AED, where angle ADE is congruent to angle BAC (α), angle DEA is congruent to angle ABC (β), and angle CED is congruent to angle BCA (γ).

The sum of the angles in triangle AED is α + β + γ.

6. Apply Euclid's fifth postulate:

According to Euclid's fifth postulate, the sum of the interior angles on one side of a straight line is less than two right angles. Since line DE is parallel to line BC, the sum of angles α, β, and γ must be less than two right angles.

7. Sum of angles in a straight line:

The sum of the angles in a straight line is equal to two right angles (180 degrees).

8. Conclusion:

Since the sum of angles α, β, and γ is less than two right angles and the sum of angles in a straight line is equal to two right angles, it follows that the sum of the three angles of a triangle (α + β + γ) is equal to two right angles, i.e., 180 degrees.