Angle Sum Property of a Triangle


Lines and Angles III - Concepts
Class - 9th CBSE 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

The sides BC, CA, and AB of a Delta ABC have been produced to D, E, and F respectively. If angle ACD=105^0 and angle EAF=45^0, find all the angles of the Delta ABC.

Right Option : B
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Explanation
Question : 2

State whether the given set of angles form a triangle

94^{0},35^{0},51^{0}

Right Option : C
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Explanation
Question : 3

If one angle of a triangle is equal to the sum of the other two angles, then the triangle is ___________________ .

Right Option : D
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Explanation
 
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