Angle Sum Triangle is a geometric name for a figure made up of three line segments intersecting at three angles. In a triangle, the total of the angles is always 180 degrees. According to the Angle Sum Triangle theorem, any triangle’s angles add up to 180 degrees. Basic geometry ideas can be used to prove this theorem. Acute, right, and obtuse angles are the three types of angles in a triangle.

The angle sum property of a triangle asserts that the sum of a triangle’s angles equals 180 degrees. Three sides and three angles, one at each vertex, make up a triangle. The total of the inner angles of any triangle, whether acute, obtuse, or right, is always 180°.

One of the most commonly utilized qualities in geometry is the angle sum property of a triangle. The most common application of this characteristic is to calculate unknown angles.

## What is the angle sum property?

The total of the three internal angles of a triangle is 180 degrees, according to the angle sum feature of a triangle. A triangle is a closed shape made up of three line segments, each of which has both interior and exterior angles. When the values of the other two angles are known, the angle sum property is utilized to compute the measure of an unknown interior angle.

### Formula for angle sum property:-

S = (n-2)*180° is the angle sum property formula for any polygon, where ‘n’ represents the number of sides in the polygon. The sum of the internal angles in a polygon may be determined using the number of triangles that can be constructed inside it, according to this polygon feature. Draw diagonals from a single vertex to make these triangles. However, a simple formula may be used to compute this, which states that if a polygon has ‘n’ sides, there will be (n – 2) triangles inside it.

Let’s use a decagon with ten sides as an example and apply the formula. S = (n-2) 180°, S = (10-2) 180° = 8*180° = 1440° are the results. As a result of the decagon’s angle sum feature, the sum of its internal angles is always 1440°. The same algorithm can be used to calculate the area of various polygons. The angle sum attribute is most commonly used to find a polygon’s unknown angles.

### The proof of the angle sum property:-

Let’s have a look at the proof of the triangle’s angle sum property. The steps for establishing a triangle’s angle sum property are outlined below:-

Step1 :- Draw a line PQ through the vertex A of the triangle ABC and parallel to side BC.

Step1 :- The total of the angles on a straight line equals 180°, as we all know. To put it another way, PAB + BAC + QAC = 180°, which gives us Equation 1: PAB + BAC + QAC = 180°.

Step1 :- Line PQ is now parallel to line BC.

∠ PAB is the same as ∠ABC, while ∠QAC is the same as ∠ACB. (Interior alternating angles), resulting in Equations 2 and 3: ∠PAB = ∠ABC and ∠QAC = ∠ACB, respectively.

Step1 :- In Equation 1, replace ∠PAB and ∠QAC with ∠ABC and ∠ACB, respectively, as illustrated below.

∠PAB + ∠BAC + ∠QAC = 180° (Equation 1). As a result, ∠ABC + ∠BAC + ∠ACB = 180°.

### Important points:-

There are fee angle sum property, that points should be kept in mind.

S = (n-2)*180° is the angle sum property formula for any polygon, where ‘n’ represents the number of sides in the polygon.

The sum of the internal angles of a polygon can be determined using the number of triangles that can be created inside it, according to the angle sum property of a polygon.

The sum of a triangle’s internal angles is always 180°.

Each angle in an equilateral triangle has a value of 60 degrees.

A right-angled triangle’s two acute angles add up to 90 degrees.

The smallest angle is opposite the smallest side, while the biggest angle is opposite the largest side.

The two angles opposite the two equal sides of a triangle are equal.

A triangle can only have one right angle or one obtuse angle at a time.

### CONCLUSION:-

A triangle is one of the most commonly used shapes in geometry. Three sides and three angles make up a triangle. The triangle’s sides and angles are its components. There are two sorts of angles in polygons: internal angles and outside angles. The triangle has three inner angles and six outside angles because it is the smallest polygon. The letters ABC stand for a triangle having the vertices A, B, and C. There are many different types of triangles, each with its own set of angles and edges, yet they always obey the triangle sum principles.

The Angle Sum Triangle Theorem is a geometric theorem that can be utilised to address triangle-related difficulties. It can be used to identify whether a triangle is acute, right, or obtuse, as well as the size of the angles of a triangle. The Angle Sum Triangle theorem can also be used to calculate the length of a triangle’s sides given the angle sizes.