ANGLE SUM PROPERTY OF A TRIANGLE

ANGLE SUM PROPERTY OF A TRIANGLE 

A triangle is a polygon that has three sides and three interior angles. It is the smallest polygon. It has three vertices. It can be an acute, obtuse or right-angled triangle. The sum of the interior angles will always remain 180 degrees. The angle sum property of a triangle is one of the most frequently used properties in geometry and is used to calculate unknown angles.

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

PROPERTIES OF A TRIANGLE

There are various properties of a triangle.

1. A triangle has three sides, three vertices and three angles.
2. The sum of all the angles inside of a triangle is always equal to 180 degrees. It does not matter which kind of a triangle it is.
3. The sum of the length of any of the two sides of a triangle will always be greater than the length of the third side.
4. The side which is opposite to the largest angle of a triangle is the largest.
5. An exterior angle (angle outside the triangle) of the triangle is equal to the sum of the opposite interior angles. This is also known as the exterior angle property of a triangle.

PROOF OF THE PROPERTY

Let us consider the triangle ABC where AB, BC and AC are sides of a triangle. ∠ABC, ∠BCA and ∠BAC are the interior angles of a triangle. 

Proof: Draw a line PQ parallel to the base of the triangle BC. 

PQ is a straight line so ∠PAB +∠BAC + ∠QAC = 180°  ——-(1)

We also know that, ∠PAB = ∠ABC and ∠QAC = ∠BCA (Since PQ is parallel to BC and AB is the transversal; Corresponding angles are equal when a transversal intersects parallel lines)

Substituting the values of ∠PAB and ∠QAC in equation (1)

∠ABC +∠BAC + ∠BCA  = 180°

Hence proved. 

Examples 

1. If in a triangle ABC, ∠A = 38°, ∠B = 96°. Then what is the value of ∠C?

Since the sum of all the angles inside a triangle  is 180° so:

∠A + ∠B + ∠C = 180°

⇒ 38° + 96° + ∠C = 180°

⇒ ∠C = 180° – 134°

  ∴ ∠C = 46°

2. In a triangle PQR, What is the value of ∠P in a triangle which is right-angled at Q and ∠R = 35°?

We know that, ∠P + ∠Q + ∠R = 180°.

∠P + 90° + 35° = 180°

⇒ ∠P = 180° – 125°

∴ ∠P = 55°