Angle Sum Property of a Triangle: Theorem, Examples and Proof (2024)

Angle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and according to this property, “Sum of All the Angles of a Triangle is equal to 180º.”

Angle Sum Property of a Triangle is applicable to any of the triangles whether it is a right, acute, obtuse angle triangle or any other type of triangle. So, let’s learn about this fundamental property of a triangle i.e., “Angle Sum Property “.

Table of Content

  • What is the Angle Sum Property?
  • Angle Sum Property Formula
  • Proof of Angle Sum Property
  • Exterior Angle Property of a Triangle Theorem
  • Angle Sum Property of Triangle Facts
  • Solved Example
  • FAQs

What is the Angle Sum Property?

For a closed polygon, the sum of all the interior angles is dependent on the sides of the polygon. In a triangle sum of all the interior angles is equal to 180 degrees. The image added below shows the triangle sum property in various triangles.

Angle Sum Property of a Triangle: Theorem, Examples and Proof (1)

This property holds true for all types of triangles such as acute, right, and obtuse-angled triangles, or any other triangle such as equilateral, isosceles, and scalene triangles. This property is very useful in finding the unknown angle of the triangle if two angles of the triangle are given.

Angle Sum Property Formula

The angle sum property formula used for any polygon is given by the expression,

Sum of Interior Angle = (n − 2) × 180°

where ‘n’ is the number of sides of the polygon.

According to this property, the sum of the interior angles of the polygon depends on how many triangles are formed inside the polygon, i.e. for 1 triangle the sum of interior angles is 1×180° for two triangles inside the polygon the sum of interior angles is 2×180° similarly for a polygon of ‘n’ sides, (n – 2) triangles are formed inside it.

Example: Find the sum of the interior angles for the pentagon.

Solution:

Pentagon has 5 sides.

So, n = 5

Thus, n – 2 = 5 – 2 = 3 triangles are formed.

Sum of Interior Angle = (n − 2) × 180°

⇒ Sum of Interior Angle = (5 − 2) × 180°

⇒ Sum of Interior Angle = 3 × 180° = 540°

Proof of Angle Sum Property

Theorem 1: The angle sum property of a triangle states that the sum of interior angles of a triangle is 180°.

Proof:

The sum of all the angles of a triangle is equal to 180°. This theorem can be proved by the below-shown figure.

Angle Sum Property of a Triangle: Theorem, Examples and Proof (2)

Follow the steps given below to prove the angle sum property in the triangle.

Step 1: Draw a line parallel to any given side of a triangle let’s make a line AB parallel to side RQ of the triangle.

Step 2: We know that sum of all the angles in a straight line is 180°. So, ∠APR + ∠RPQ + ∠BPQ = 180°

Step 3: In the given figure as we can see that side AB is parallel to RQ and RP, and QP act as a transversal. So we can see that angle ∠APR = ∠PRQ and ∠BPQ = ∠PQR by the property of alternate interior angles we have studied above.

From step 2 and step 3,

∠PRQ + ∠RPQ + ∠PQR = 180° [Hence Prooved]

Example: In the given triangle PQR if the given is ∠PQR = 30°, ∠QRP = 70°then find the unknown ∠RPQ

Solution:

As we know that, sum of all the angle of triangle is 180°

∠PQR + ∠QRP + ∠RPQ = 180°

⇒ 30° + 70° + ∠RPQ = 180°

⇒ 100° + ∠RPQ = 180°

⇒ ∠RPQ = 180° – 100°

⇒ ∠RPQ = 80°

Exterior Angle Property of a Triangle Theorem

Theorem 2: If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.

Proof:

Angle Sum Property of a Triangle: Theorem, Examples and Proof (3)

As we have proved the sum of all the interior angles of a triangle is 180° (∠ACB + ∠ABC + ∠BAC = 180°) and we can also see in figure, that ∠ACB + ∠ACD = 180° due to the straight line. By the above two equations, we can conclude that

∠ACD = 180° – ∠ACB

⇒ ∠ACD = 180° – (180° – ∠ABC – ∠CAB)

⇒ ∠ACD = ∠ABC + ∠CAB

Hence proved that If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.

Example: In the triangle ABC, ∠BAC = 60° and ∠ABC = 70° then find the measure of angle ∠ACB.

Solution:

The solution to this problem can be approached in two ways:

Method 1: By angle sum property of a triangle we know ∠ACB + ∠ABC + ∠BAC = 180°

So therefore ∠ACB = 180° – ∠ABC – ∠BAC

⇒ ∠ACB = 180° – 70° – 60°

⇒ ∠ACB = 50°

And ∠ACB and ∠ACD are linear pair of angles,

⇒ ∠ACB + ∠ACD = 180°

⇒ ∠ACD = 180° – ∠ACB = 180° – 50° = 130°

Method 2: By exterior angle sum property of a triangle, we know that ∠ACD = ∠BAC + ∠ABC

∠ACD = 70° + 60°

⇒ ∠ACD = 130°

⇒ ∠ACB = 180° – ∠ACD

⇒ ∠ACB = 180° – 130°

⇒ ∠ACB = 50°

Read More about Exterior Angle Theorem.

Angle Sum Property of Triangle Facts

Various interesting facts related to the angle sum property of the triangles are,

  • Angle sum property theorem holds true for all the triangles.
  • Sum of the all the exterior angles of the triangle is 360 degrees.
  • In a triangle sum of any two sides is always greater than equal to the third side.
  • A rectangle and square can be divided into two congruent triangles by their diagonal.

Also, Check

  • Area of a Triangle
  • Area of Isosceles Triangle

Solved Example on Angle Sum Property of a Triangle

Example 1: It is given that a transversal line cuts a pair of parallel lines and the ∠1: ∠2 = 4: 5 as shown in figure 9. Find the measure of the ∠3?

Angle Sum Property of a Triangle: Theorem, Examples and Proof (4)

Solution:

As we are given that the given pair of a line are parallel so we can see that ∠1 and ∠2 are consecutive interior angles and we have already studied that consecutive interior angles are supplementary.

Therefore let us assume the measure of ∠1 as ‘4x’ therefore ∠2 would be ‘5x’

Given, ∠1 : ∠2 = 4 : 5.

∠1 + ∠2 = 180°

⇒ 4x + 5x = 180°

⇒ 9x = 180°

⇒ x = 20°

Therefore ∠1 = 4x = 4 × 20° = 80° and ∠2 = 5x = 5 × 20° = 100°.

As we can clearly see in the figure that ∠3 and ∠2 are alternate interior angles so ∠3 = ∠2

∠3 = 100°.

Example 2: As shown in Figure below angle APQ=120° and angle QRB=110°. Find the measure of the angle PQR given that the line AP is parallel to line RB.

Angle Sum Property of a Triangle: Theorem, Examples and Proof (5)

Solution:

As we are given that line AP is parallel to line RB

We know that the line perpendicular to one would surely be perpendicular to the other. So let us make a line perpendicular to both the parallel line as shown in the picture.

Now as we can clearly see that

∠APM + ∠MPQ = 120° and as PM is perpendicular to line AP so ∠APM = 90° therefore,

⇒ ∠MPQ = 120° – 90° = 30°.

Similarly, we can see that ∠ORB = 90° as OR is perpendicular to line RB therefore,

∠QRO = 110° – 90° = 20°.

Line OR is parallel to line QN and MP therefore,

∠PQN = ∠MPQ as they both are alternate interior angles. Similarly,

⇒ ∠NQR = ∠ORQ

Thus, ∠PQR = ∠PQN + ∠NQR

⇒ ∠PQR = 30° + 20°

⇒ ∠PQR = 50°

FAQs on Angle Sum Property

Define Angle Sum Property of a Triangle.

Angle Sum Property of a triangle states that the sum of all the interior angles of a triangle is equal to 180°. For example, In a triangle PQR, ∠P + ∠Q + ∠R = 180°.

What is the Angle Sum Property of a Polygon?

The angle sum property of a Polygon states that for any polygon with side ‘n’ the sum of all its interior angle is given by,

Sum of all the interior angles of a polygon (side n) = (n-2) × 180°

What is the use of the angle sum property?

The angle sum property of a triangle is used to find the unknown angle of a triangle when two angles are given.

Who discovered the angle sum property of a triangle?

The proof for triangle sum property was first published by, Bernhard Friedrich Thibaut in the second edition of his Grundriss der Reinen Mathematik

What is the Angle Sum Property of a Hexagon?

Angle sum property of a hexagon, states that the sum of all the interior angles of a hexagon is 720°.



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Angle Sum Property of a Triangle: Theorem, Examples and Proof (2024)

FAQs

Angle Sum Property of a Triangle: Theorem, Examples and Proof? ›

We add the two known angles and subtract their sum from 180° to get the measure of the third angle. For example, if two angles of a triangle are 70° and 60°, we will add these, 70 + 60 = 130°, and we will subtract it from 180°, which is the sum of the angles of a triangle. So, the third angle = 180° - 130° = 50°.

How to prove the triangle angle sum theorem? ›

We can draw a line parallel to the base of any triangle through its third vertex. Then we use transversals, vertical angles, and corresponding angles to rearrange those angle measures into a straight line, proving that they must add up to 180°.

What is the proof of the angle angle theorem? ›

Angle Angle Side Congruence Theorem

If both the triangles are superimposed on each other, we see that ∠B =∠E and ∠C =∠F. And the non-included sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.

What is the proof theorem of triangles? ›

What are the triangle proofs? Triangle proofs are five different ways to prove if two triangles are the same size and shape. These five triangle congruencies are SSS, SAS, ASA, AAS, and HL.

What is an example of the angle sum property of a triangle? ›

Angle Sum Property of a triangle states that the sum of all the interior angles of a triangle is equal to 180°. For example, In a triangle PQR, ∠P + ∠Q + ∠R = 180°.

What is the angle sum formula? ›

The sum of the interior angles of a given polygon = (n − 2) × 180°, where n = the number of sides of the polygon.

What is the angle theorem proof? ›

The same side interior angles theorem describes an angles proof according to the statement: 'If two parallel straight segments A and B are crossed by a transversal segment C, two adjacent interior angles are supplementary (sum 180 degrees). '

What is the triangle theorem for angle? ›

As per the triangle sum theorem, the sum of all the angles (interior) of a triangle is 180 degrees, and the measure of the exterior angle of a triangle equals the sum of its two opposite interior angles.

What is the angle theorem rule? ›

A polygon is equiangular if the interior angles are congruent to each other and equilateral if the edges are congruent. An equiangular and equilateral polygon is called a regular polygon. The interior angle theorem states that the sum of the interior angles of a polygon with n vertices is S n = 180 ( n − 2 ) .

How do you verify that the sum of three angles of a triangle is 180 degrees by paper cutting and pasting? ›

Now, draw a line on the cardboard and paste the cut outs of the angles (∠A, ∠B and ∠C) on the line at a point O as shown in Fig. 12.4. When all the three cut outs of the angles A, B, C placed adjacent to each other at a point, then it forms a line forming a straight angle, i.e. 180°.

What is the exterior angle sum theorem? ›

What is the Exterior Angle Theorem? The exterior angle theorem states that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles of the triangle. The remote interior angles are also called opposite interior angles.

What are 5 ways to prove a triangle? ›

There are five theorems that can be used to show that two triangles are congruent: the Side-Side-Side (SSS) theorem, the Side-Angle-Side (SAS) theorem, the Angle-Angle-Side (AAS) theorem, the Angle-Side-Angle (ASA) theorem, and the Hypotenuse-Leg (HL) theorem.

How do you write a proof for a theorem? ›

The Proof-Writing Process

A proof must always begin with an initial statement of what it is you intend to prove. It should not be phrased as a textbook question (“Prove that….”); rather, the initial statement should be phrased as a theorem or proposition.

What is the formula of triangle proof? ›

The formula for that is 1/2*b*h, where b is base length and h is height. Hope this helped.

How do you prove the angle angle side theorem? ›

In order to use AAS, all that is necessary is identifying two equal angles in a triangle, then finding a third side adjacent to only one of the angles in each of the triangles such that the two sides are equal. This is enough to prove the two triangles are congruent.

How do you prove the base angle theorem? ›

To prove the Base Angles Theorem, we will construct the angle bisector through the vertex angle of an isosceles triangle. By constructing the angle bisector, E G ¯ , we designed two congruent triangles and then used CPCTC to show that the base angles are congruent.

How do you prove angle addition postulates? ›

The formula of angle addition postulate in math is used to express the sum of two adjacent angles. If there are two angles (∠AOB and ∠BOC) joined together sharing a common arm OB and a common vertex O, then the angle addition postulate formula is ∠AOB + ∠BOC = ∠AOC.

How do you prove a triangle formula? ›

Proof of Right Angle Triangle Theorem
  1. Theorem:In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
  2. To prove: ∠B = 90°
  3. Proof: We have a Δ ABC in which AC2 = AB2 + BC2
  4. Also, read:
  5. c2 = a2 + b2
  6. c = √(a2 + b2)
  7. A = 1/2 b x h.

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