Introduction to the triangle sum conjecture:
The triangle sum conjecture is nothing but the triangle angle sum theorm. The word conjecture means theorm. The sum of all the interior angles are equal to 180^{o}. If one side of a triangle is produced, the exterior angle created will be equal to the summation of the interior opposite angles. Here we are going to see about the triangle sum conjecture.
Triangle sum conjecture is equal to two right angles, i.e., 180 degrees
Given:
ABC is a triangle
To Prove
Angle A + Angle B + Angle ACB = 180^{o}
Produce BC to D. From C we draw CE || BA.
Proof for triangle sum conjecture
Statement |
Reason |
1. Angle A = Angle ACE |
Alternate angles angles BA is parallel to CE |
2. Angle B = Angle ECD |
Here Corresponding angles BA is parallel to CE |
3. Angle A + angle B = Angle ACE + Angle ECD |
statements (1) and (2) |
4. Angle A + angle B = Angle ACD |
statement (3) |
5. Angle A + Angle B + Angle ACB = Angle ACD + Angle ACB |
adding Angle ACB to both sides |
6. But Angle ACD + Angle ACB = 180^{o} |
linear pair |
7. Angle A + Angle B + Angle ACB = 180 ° |
statements (5) and (6) |
If one side of a triangle is formed then the exterior angle so formed is equal to the sum of the interior opposite angles.
Given:
In the given triangle ABC, BC is produced to D.
To Prove triangle sum conjecture:
Angle ACD = Angle A + Angle B
Proof:
Statement |
Reason |
1. Angle ACB + Angle ACD = 180^{o}. |
It is a linear pair |
2. Angle A + Angle B + Angle ACB = 180^{o} |
sum of the angles of a triangle = 180 |
3. Angle ACB + Angle ACD = Angle A + Angle B + Angle ACB |
statements (1) and (2) |
4. Angle ACD = Angle A + Angle B Reason |
statement (3); Angle ACB is common |