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 180o.  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:

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 = 180o

Produce BC to D. From C we draw CE || BA.

angle sum theorem

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 = 180o

linear pair

7. Angle A + Angle B + Angle ACB = 180 °

statements (5) and (6)

Corollary for Triangle Sum Conjecture:

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

Corollary of Sum of Angles of Triangle

Proof:

Statement

Reason

1. Angle ACB + Angle ACD = 180o.

It is a linear pair

2. Angle A + Angle B + Angle ACB = 180o

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