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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

A triangle cannot have more than ______________ right angles.
A. 1	B. 2	C. 3	D. 0
Right Option : A
View Explanation

Question : 2

If $dpi{100} EC \|\| AB$ , $angle ECD=70^0$ and $angle BDO=20^0$ , then $angle OBD$ is
A. $20^0$	B. $50^0$	C. $60^0$	D. $70^0$
Right Option : B
View Explanation

Question : 3

Find the measure of angle U in the following diagram
A. $80^{0}$	B. $75^{0}$	C. $60^{0}$	D. $50^{0}$
Right Option : A
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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