An Introduction To Angle Sum Property Of A Triangle

The Angle Sum Property of a triangle is when three line segments intersect at three angles, and the sum of the angles results in 180. Triangle consists of three vertices and three angles which are also known as the elements of the triangle. The Angle Sum Property is one of the most widely used properties in geometry which is used to calculate the size, type, and length of the sides of the triangle.

Key Terms: Triangle, Angle Sum Property, Interior Angles, Right-angled Triangle, Straight Line, Angles

---

Proof of Angle Sum Property

[Click Here for Sample Questions]

Angle Sum Property

The following steps prove the Angle Sum Property theorem:

Step 1: Draw a straight line AB which should be parallel to the triangle’s side XZ. The line AB Should pass through the vertex Y.

Step 2: ∠BYZ + ∠XYZ + ∠ZYA = 180°

the sum of angles on the straight line = 180°

Step 3: ∠BYZ = ∠YXZ

Alternate Interior Angles

Step 4: ∠AYX = ∠YZX 

Step 5: By Substitution,

∠BYZ +∠XYZ + ∠AYZ = 180°

∠BYZ = ∠YXZ (AB //XZ and XY are transversal)

∠AYZ = ∠YZX (AB //XZ and YZ are transversal)

Hence proved, in triangle XYZ, ∠ YXZ + ∠ XZY + ∠ZYX = 180°

Thus, the sum of the interior angles of a triangle is 180°

Angle Sum Property

Read More:

---

Properties of a Triangle

[Click Here for Sample Questions]

• The Angle Sum Property states that ∠A + ∠B +∠C = 180° in a triangle whose all the sides are equal (equilateral triangle), the value of each side is 60°.
• A Triangle consists of one obtuse angle or one right angle at the most.
• When two sides are equal, the two angles of the triangles that are formed opposite are also equal.
• The smallest angle is formed opposite the smallest side and the largest angle is formed opposite the largest side. 
• In a Right-Angle Triangle, the sum of two acute angles is 90°.

---

Triangle Sum Property

[Click Here for Sample Questions]

[Click Here for Sample Questions]

A Triangle consists of three angles and sides along with a line Segment, Interior Angles, and Exterior Angles. There are three Interiors and six Exterior Angles. A Triangle can have different shapes and sizes, but the total Interior Angle is always 180°.

---

Angle Sum Property of a Triangle theorem

[Click Here for Sample Questions]

The vertex of any two edges of a triangle joins to form interior angles. The Interior angle is the angle formed between two sides of the triangle. The formula of the Angle Sum Property of a triangle is

Interior Angle Sum Property of Triangle

Theorem 1:

Triangle angle sum property indicates that the sum of a triangle's inner angles is 180°.

To prove: ∠A + ∠B + ∠C = 180°

Construction: Draw a line passing through point A that is parallel to side BC of the given triangle.

Proof: Assuming PQ is a straight line, we may conclude from the linear pair that: ∠1 + ∠2 + ∠3 = 180°......... (1)

Because PQ || BC and AB, AC are transversals,

As a result, 3 =∠ACB (a pair of alternate angles)

In addition, 1 = ∠ABC (a pair of alternate angles)

By changing the values of 3 and 1 in equation (1), ∠ABC + ∠BAC + ∠ACB

= 180° ∠A + ∠B +∠C = 180° 

⇒∠A + ∠B + ∠C = 180° = 2 x 90° = 2 right angles

Exterior Angle Property of Triangle

Theorem 2:

As a result, the total of a triangle's internal angles is 180°.

If any one side of a triangle is constructed, the exterior angle formed is equal to the sum of two opposite interior angles.

Consider the following triangle ABC, whose side BC is extended D to generate an exterior angle ∠ACD.

To show ∠ACD = ∠BAC + ∠ABC or ∠4 = ∠1 + ∠2

Proof: ∠3 and ∠4 forms a linear pair since they represent adjacent straight-line angles.

Thus, ∠3 + ∠4 = 180° ……….(2) 

Furthermore, the interior angle sum condition of triangles dictates that: ∠1 + ∠2 + ∠3 = 180°........... (3)

According to equations (2) and (3), ∠4 = ∠1 + ∠2 

⇒∠ACD = ∠BAC + ∠ABC

As a result, a triangle's exterior angle equals the sum of its opposing interior angles.

---

Angle Sum Property Formula

[Click Here for Sample Questions]

S = (n – 2)180° is the angle sum property formula for any polygon, where 'n' indicates 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 property. 

Draw diagonals from a single vertex to construct these triangles. To make things easier, 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.

Things to Remember

• The sum of the Interior Angles of a triangle is 180°
• The Angle Sum Property of a triangle is also known as the Interior Angle Property of a triangle. 
• The Angle Sum Property of any Polygon (S) = (n-2) x 180° n = number of sides in the Polygon.
• Angle sum property of a triangle can be used to determine whether a given shape is a triangle or to determine the missing angle in a triangle.
• Triangles are categorised based on: Triangles with acute, obtuse, and right angles are examples of interior angles. The length of the sides of an equilateral triangle, an isosceles triangle, and a scalene triangle.

Also Read: