Angle Sum Triangle

What is a Triangle?

In our daily life, we all see a lot of things that are of different shapes and structures. Things like Traffic signs, pyramids, truss bridges, sailboards, roof a house area few things that look like a triangle. So what is a triangle? A triangle is a closed polygon that is formed by only three line segments. Triangles can be classified on the basis of their sides and angles. On the basis of their sides, triangles can be an equilateral, isosceles, and scalene. And on the basis of their angle, a triangle can be an acute triangle, right triangle, an obtuse triangle. 

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Properties of Triangles

The properties of a triangle are listed below

  • The sum of all the internal angles of a triangle is always equal to  180 degrees.

  • The sum of the length of any two sides of a triangle is always greater than the third side of the triangle.

  • The difference between the length of any two sides of a triangle is always less than the length of the third side of the triangle.

  • The side that is at the opposite side of the greater angle is always the longest side among all the three sides of a triangle.

  • In a triangle, the exterior angle is always equal to the sum of the interior opposite angle. This property is known as exterior angle property. 

  • Any two triangles will be similar if their corresponding angles tend to be congruent and length of their sides will be proportional.  

  • The area of a triangle is ½ x base x height

  • The perimeter of the triangle is the sum of all its three sides. 


Interior and Exterior Angle of a Triangle

There are two important attributes of a triangle i.e., angle sum property and exterior angle property. We may have questions in our mind about what is an angle sum property of a triangle? And how can we prove angle sum property? So lets clear our head with these questions. 

We know already that a triangle has an interior as well as an exterior angle. The Interior angle is an angle between the adjacent sides of a triangle and an exterior angle is an angle between the side of a triangle and an adjacent side extending outward.


Angle Sum Property

Theorem: Prove that the sum of all the three angles of a triangle is 180 degrees or 2 right angles. 

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Proof:  ∠1 = ∠B and ∠3 = ∠C………….(i)

Alternative angle = PQ||BC

∠1 +∠2 +∠3 = 180

∠B +∠2 + ∠C = 180

∠B +∠CAB + ∠C = 180

= 2 right angles 

Proved 

Theorem 2: In a triangle, if one side is formed then the exterior angle formed will be equal to the sum of two interior opposite angles.

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 ∠4 = ∠1 + ∠2

Proof: ∠3 = 180 - (∠1 +∠2)..............(i)

∠3 + ∠4 = 180

Or ∠3 = 180 - ∠4…………(ii)

By (i) and (ii)

180 - (∠1 + ∠2) = 180 - ∠4

∠1 + ∠2 = ∠4

Proved


Exterior Angle Property

Proof of exterior angle property

The exterior angle theorem asserts that if a triangle’s side gets extended, then the resultant exterior angle will be equal to the total of the two opposite interior angles of the triangle.

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According to the Exterior Angle Theorem, the sum of measures of ∠ABC and ∠CAB will be equal to the exterior angle ∠ACD. The general proof of this theorem is explained below:

Proof:

Consider  ∆ABC as given below such that the side BC of ∆ABC is extended. A parallel line to the side AB is drawn.

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Serial no   

Statement 

Reason 

1

∠CAB = ∠ACE

⇒∠1=∠x

Pair of alternate angles 

((BA)॥(CE))and(AC) is the transversal)

2

∠ABC = ∠ECD

⇒∠2 = ∠y

Corresponding angles ((BA)॥(CE))and(BD)

 is the transversal) 

3

⇒∠1+∠2 = ∠x+∠y

From statements 1 and 2

4

∠x+∠y = ∠ACD

From fig. 3

5

∠1+∠2 = ∠ACD

From statements 3 and 4


Thus, from the above statements, we can see that the exterior angle ∠ACD of ∆ABC is equal to the sum of two opposite interior angles i.e. ∠CAB and ∠ABC of the ∆ABC.

Solved Examples

Example 1) In the following triangle, find the value of x.

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Solution 1) x + 24 + 32 = 180

                  x + 56 = 180

                  x = 180 - 56

                  x = 124

Example 2) In the triangle ABC given below, find the area of a triangle that inscribed inside a square of 20 cm.

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Solution 2) Area of a triangle = ½ x base x height

                                               = ½ (20)(20)

                                               = 200 cm²

FAQ (Frequently Asked Questions)

Question 1. How to Find the Missing Angle of a Triangle?

Answer. According to the general rule, we know that the sum of all the angles of a triangle will always add up to 180 degrees. So even if one angle of the triangle is unknown to us and the two angles are known, we can still find the unknown angle of a triangle. What we have to do to find the unknown angle is to add the two known angles and then subtract the sum of the two angles with 180 degrees. For example, in a triangle, two angles measure 72 degrees and 28 degrees respectively and 1 angle is unknown. We can find the unknown angle if we add the two known angles and then subtract it from 180 degrees. 

        72 + 28 = 100

        180 - 100 = 80  

Therefore, the unknown angle or the third angle of the triangle will measure 80 degrees. 

Question 2. How to Find the Missing Length of the Right Angle Triangle?

Answer. Just like the missing angle of a triangle can be found, we can also find the missing length of a right angle triangle. Also, the same condition will be applied that the two other lengths of the right angle triangle are known. We can use the Pythagoras theorem and find the third length of the triangle. Pythagoras theorem states that the square of hypothenuse is equal to the squared sum of the other two sides h2=a2+b2

In another case, if we know one side and one angle of a triangle, then we can take the help of trigonometry ratio i.e., sine, cosine, and tangent.