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Angle Sum Property of a Triangle

Lines and Angles III - Concepts

Angle Sum Property of a Triangle

Exterior Angle Property

Class - 9th IMO Subjects

Concept Explanation

Angle Sum Property of a Triangle

Theorem: The sum of the three angles of a triangle is $large 180^{0}$

Given: A triangle ABC.

To Prove: $large angle A+angle B+angle C=180^{0}$ i.e. $large angle 1+angle 2+angle 3=180^{0}$

Construction: Through A, draw a line l parallel to BC.

Proof: Since $large lparallel BC$ . Therefore,

$large angle 2=angle 4$ [Alternate interior angles] .................(1)

and, $large angle 3=angle 5$ [Alternate interior angles] .................(2)

Adding equations (1),(2), we have

$large therefore ;;;;angle 2+angle 3=angle 4+angle 5$

$large Rightarrow ;;;angle 1+angle 2+angle 3=angle 1+angle 4+angle 5$ [Adding $large angle 1$ on both sides]

$large Rightarrow ;;;angle 1+angle 2+angle 3=angle 4+angle 1+angle 5$

$large Rightarrow ;angle 1+angle 2+angle 3=180^{0}$ [ $large because$ Sum of angles at a point on a line is $large 180^{circ}$

$large therefore$ $large angle 4+angle 1+angle 5=180^{circ}$ ]

$large Rightarrow;angle A+angle B+angle C=180^{0}$

Thus, the sum of the three angles of a triangle is $large 180^{0}$

Illustration: In a $bigtriangleup ABC,angle B=115^{0},angle C=45^{0}.$ Find $angle A$

Solution: Given: In $bigtriangleup ABC,angle B=115^{0},angle C=45^{0}.$ ...............(1)

By the angle sum property of a triangle, we have

$large angle A+angle B+angle C=180^{0}$

Using the values from equation (1), we have

$large angle A+115^{0}+45^{0}=180^{0}$

$large angle A+160^{0}=180^{0}$

$large angle A=180^{0}-160^{0}$

$large angle A=20^{0}$

Illustration: A, B, C are the three angles of a triangle. If $large A-B=15^{0},B-C=30^{0},$ Find $large angle A,angle B;and;angle C$ .

Solution: Given: $large A-B=15^{0}$ ..................(1)

$large A= B+15^{0}$ ..................(2)

and $large B-C=30^{0}$ ..................(3)

$large B= C+30^{0}$ ..................(4)

$large C= B-30^{0}$ ..................(5)

Using the angle sum property of a triangle

$large angle A+angle B+angle C=180^{0}$

Using equations (2),(5), we have

$large B+15^{0}+B+B-30^{0}=180^{0}$

$large 3B-15^{0}=180^{0}$

$large 3B=180^{0}+15^{0}$

$large 3B=195^{0}$

$large B=frac{195^{0}}{3}$

$large B=65^{0}$ .....................(6)

Using equation (6) in equations (2),(5), we have

$large A= B+15^{0}=65^{0}+15^{0}=80^{0}$

$large C= B-30^{0}=65^{0}-30^{0}=35^{0}$

Hence $large A=80^{0},;B=65^{0;},C=35^{0}$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

State whether the given pair of angles form a triangle $52^{0},83^{0},45^{0}$
A. Yes if angle $45^{0}$ is changed to $55^{0}$	B. Yes if angle $45^{0}$ is changed to $35^{0}$	C. No	D. Yes it will form a triangle
Right Option : D
View Explanation

Question : 2

Find the value of x:
A. $large dpi{120} fn_jvn large 33^0$	B. $large 29^0$	C. $large 48^0$	D. $large 17^0$
Right Option : A
View Explanation

Question : 3

Find the value of x:
A. $large dpi{120} fn_jvn large -10^0$	B. $large 20^0$	C. $large -11^0$	D. $large 11^0$
Right Option : C
View Explanation

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Language - English

Chapters

Polynomials II

Direct and Inverse Variation

Linear Equations in One Variable

Introduction to Trigonometry

Introduction to Euclids Geometry

Herons Formula Complete

Polynomial I

Surface Area and Volume II

Surface Area and Volume I

Coordinate Geometry

Constructions

Areas Of Parallelograms And Triangles II

Areas Of Parallelograms And Triangles I

Linear Equations in Two Variables

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Attention !!!!!

Angle Sum Property of a Triangle

Angle Sum Property of a Triangle

Students / Parents Reviews [20]

Aman Kumar Shrivastava

Shreya Shrivastava

Yogansh Nyasi

Manish Kumar

Aman Kumar Shrivastava

Barkha Arora

Hridey Preet

Anika Saxena

Manish Kumar

Ayan Ghosh

Prisha Gupta

Hiya Gupta

Shivam Rana

Aman Kumar Shrivastava

Ayan Ghosh

Archit Segal

Yatharthi Sharma

Cheshta

Om Umang

Shreya Shrivastava