Introduction to Angle Sum Triangle

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Triangle Sum Property

In geometry, one of the most used shapes is a triangle. A triangle has three sides and three angles. These sides and angles are the elements of the triangle. All the polygons have two types of angles which are interior angles and exterior angles. As the triangle is the smallest polygon, it has three interior angles and six exterior angles. A triangle with vertices A, B, C is denoted by ∆ABC. There are various kinds of triangles with different angles and edges, but, all of them follow the triangle sum properties. The two most important properties are the angle sum property of a triangle and the exterior angle property of a triangle.


Angle Sum Property of a Triangle

This property states the sum of the interior angles of a triangle is 180 degrees. Interior angles are formed at the vertex where any two edges of a triangle join. The angle between two sides of a triangle is called the interior angle. It is also known as the interior angle property of a triangle. This property states that the sum of all the interior angles of a triangle is 180°. If the triangle is ∆ABC, the angle sum property formula is ∠A+∠B+∠C = 180°.


Proof of Angle Sum Property of a Triangle

The triangle is denoted as ∆ABC. To prove the angle sum property of a triangle, draw a straight-line PQ. PQ is passing through A and parallel to BC.

∠PAB + ∠BAC + ∠QAC = 180°….. (1)

Since PQ||BC and AB & AC are transversals,

∠QAC = ∠ACB (they are alternate angles)

∠PAB = ∠CBA (they are alternate angles)

Putting the value of ∠PAB and ∠QAC in equation (1), we get

∠ABC + ∠BAC + ∠ACB = 180°

Hence, the sum of all the interior angles of a triangle is 180° (proved).


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Exterior Angle Property of a Triangle

Another vital property of the triangle is the exterior angle property. One side of a triangle and the extended the adjacent side form an exterior angle. This property gives the measurement concept of exterior angles. The exterior angle property states that the exterior angle value is equal to the sum of the two opposite interior angles of the triangle. Side BC of the triangle ∆ABC is extended up to D. The exterior angle ∠ACD is equal to the sum of the two opposite interior angles ∠ABC and ∠BAC, which means ∠ABC + ∠BAC = ∠ACD.


Proof of Exterior Angle Property of a Triangle

Here, the triangle is ∆ABC. The BC side is extended up to D.

As BD is a straight line, ∠ABC and ∠ACD form an adjacent pair on BD.

So, ∠ABC + ∠ACD = 180°…… (1)

Following the angle sum property of a triangle, we can say that

∠ABC + ∠ACB + ∠BAC = 180°...…. (2)

From equation (1) & (2), we get

∠ACD = ∠ABC + ∠BAC

Hence, the exterior angle property of a triangle is proved.


Some Other Important Angle Properties of a Triangle

Besides the angle sum property and the exterior angle property of a triangle, there are some other essential properties of the angles of a triangle, and they are as follows.

  • The value of each angle of an equilateral triangle is 60°.

  • The sum of the two acute angles of a right-angled triangle is 90°.

  • The angle opposite to the smallest side is the smallest, and the largest angle is the opposite to the largest side.

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

  • A triangle has a maximum of one right angle or one obtuse angle.

Solved Examples

1. Find Out the Angle ∠ABC of the Triangle ∆ABC. The Exterior ∠ACD = 125° and the Other Interior Angle ∠BAC = 61°.

Ans: BC ia side of ∆ABC is extended up to D, and the exterior angle is 125°. So, the two opposite angles are ∠ABC and ∠BAC. The sum of the two angles is equal to the value of ∠ACD = 125°.

Therefore, ∠ABC = ∠ACD – ∠BAC

= 125° – 61°

= 64°


2. The Ratio of the Three Angles of a Triangle is 1:2:3. Determine the Largest Angle of the Triangle and the Type of the Triangle.

Ans: According to the angle sum property, 

x + 2x + 3x = 180°

3x = 90°

Therefore, the largest angle is 90°, and it is a right-angled triangle.

FAQ (Frequently Asked Questions)

1. How to Find Exterior Angles of a Triangle?

Ans: Exterior angle is the angle between one side of the triangle and the extended adjacent side. A triangle has six exterior angles. Each edge of a triangle can form two exterior angles with the two of it's extended adjacent sides. If you want to get an exterior angle of a triangle, you have to extend the straight line of one side of the triangle. The straight line can be extended to any of the two sides. After the extension, the extended part and the adjacent side of the triangle form an angle. Thus, the exterior angle of a triangle is formed. The exterior angles have some properties as well.

2. What are the Principal Properties of the Angles of a Triangle?

Ans: The angles of a triangle are of two types, which are interior angles and exterior angles. The main two properties for these two angles of the triangle are angle sum property and exterior angle property. Apart from these two, there are many properties of the angle of a triangle. The equality of angles of an equilateral triangle comes under interior angle properties. The exterior angle property is that the exterior angle adjacent to the largest exterior angle measures the smallest. The angles of a triangle opposite to the equal sides are equal. A triangle cannot have two or more right angles.