Introduction to Angle Sum Triangle – Property, Proof and Solved Examples

Triangle Sum Property

The Triangle Sum Property states that the sum of the measures of the angles of a triangle is 180 degrees. This is because the angles of a triangle always add up to 180 degrees.

Angle Sum Property of a Triangle

The angle sum property of a triangle states that the sum of the angles in a triangle is 180 degrees. This property can be proven using basic geometry.

Proof

The triangle sum property, also known as the angle sum property of a triangle, states that the sum of the interior angles of a triangle is equal to 180 degrees. This property can be proved using Euclidean geometry as follows:

Consider an arbitrary triangle ABC with three vertices A, B, and C. Draw a line segment CD parallel to side AB, where C is a point on side BC and D is a point on side AC. This divides the triangle ABC into two smaller triangles, ACD and CBD.

Now, we know that the sum of the interior angles of a line is 180 degrees. Therefore, the sum of the angles ADC and BCD is 180 degrees.

Next, we know that CD is parallel to AB, which means that the alternate interior angles are congruent. Therefore, angle ACD is congruent to angle B.

Similarly, angle BCD is congruent to angle A.

Therefore, we can conclude that the sum of the angles of triangle ABC is:

angle A + angle B + angle C = angle A + angle ACD + angle BCD

= angle A + angle B + 180 degrees – (angle A + angle B)

= angle C + 180 degrees – angle C

= 180 degrees

Thus, we have proved that the sum of the angles of any triangle is equal to 180 degrees, which is the triangle sum property.

Some other Important Angle Properties of a Triangle

• The Interior Angle Sum of a Triangle is 180 degrees.
• The exterior angle of a triangle is the sum of the two opposite interior angles.
• The angle between two lines is equal to the angle between the lines’ projections onto a third line.