Table of Content
In this particular article, the topic of the Angle Sum Triangle has been explained in great detail using simple language and examples by the subject experts at Vedantu. Students are advised to download the free PDF of the notes and can also watch the video lectures on the topic to get a complete understanding and excel in their performance in any examination.
What is a Triangle?
Properties of Triangle
Interior and Exterior Angle of a Triangle
Angle sum property
Exterior Angle property
Solved examples
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, and roofing a house are a 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 based on their sides and angles. Based on their sides, triangles can be equilateral, isosceles, and scalene. And based on their angle, a triangle can be an acute triangle, right triangle, an obtuse triangle.
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 the 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 minds about what is an angle sum property of a triangle? And how can we prove angle sum property? So let's clear our heads 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.
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.
∠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.
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.
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.
Solution)
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 inscribed inside a square of 20 cm.
Solution) Area of a triangle
= \[\frac{1}{2} \times base \times height\]
= \[\frac{1}{2} (20) (20) \]
= \[200 cm^{2}\]
Key Learnings from the Chapter -
Students should be able to identify and label the three interior angles of any triangle
Remember that some of the interior angles of any triangle are equal to 180 degrees
There are two methods with which students can calculate the interior of any triangle
FAQs on Angle Sum Triangle
1. How to Find the Missing Angle of a Triangle?
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.
2. How to Find the Missing Length of the Right Angle Triangle?
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 the 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.
3. Can we find the missing angle of a triangle?
In order to find the missing angle of a triangle students first need to identify and label the two angles. Now you have to be certain that all triangles have interior angles adding to 180 degrees. After which you can choose any of the methods given below -
You can subtract the two known angles from 180 degrees
Or you can plug the two angles into the formula and use algebra:
a + b + c = 180 degrees
To understand the concept well students can watch the video lectures where faculties have explained the concept in the simplest way.
4. Why is the interior sum of a triangle always 180 degrees?
Imagine or draw a triangle whose interior angles are a, b, and c, and their corresponding exterior angles are a’, b’, and c’.
So,
a + a’ =180
b + b’ = 180
c + c’ = 180
Here, the sum of the external angles of a convex polygon at each vertex is 360
i.e., a’ + b’ + c’ = 360
And, (a + a’)+(b + b’)+(c + c’) = 180 +180 + 180
a + b + c + (a’ + b’ + c’) = 540
a + b + c + 360 = 540
a + b + c = 180
Therefore, the sum of the measures of the interior angle is always 180 degrees.
5. What are the types of triangles?
There are three types of basic triangles -
Equilateral triangle - where all the three sides have the same length
Isosceles triangle - where the two sides have the same length and one side has a different length
Scalene triangle - where all the three sides have different length
The shape of the triangle has an impact on the angle of the triangle. Thus, to find the angle of a triangle you must first be clear about the type of the triangle.
6. How can I score full marks in the topic Angle Sum of Triangle?
The topic Angle Sum of Triangle is a part of basic geometry and the question papers have given good weightage to the topic. And the goal of achieving full marks from the topic is quite achievable, students just need to follow a few steps with utmost sincerity -
First, try to get conceptual clarity in the topic from the available study materials and video lectures on the website of Vedantu
Once you know the topic well, start solving questions related to the topic from the available set of sample papers on Vedantu’s website
Compare your answers with the solutions provided by subject experts
Marks your mistakes and learn the new concepts
Revise the topic again
With this regular and rigorous practice, students can surely achieve the highest scores in any examination.
7. What are the other shapes where the angle sum property can be used?
Apart from the angle sum of the triangle, the angle sum property can be used for -
Angle sum of a Polygon
Angle sum of a Hexagon
Angle sum of a Quadrilateral
Angle sum of Pentagon and so on.