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Concept Detail
Maths / Triangles / Exterior Angle Property
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Exterior Angle Property

Exterior Angle: If the side BC of a triangle ABC is produced to form ray BD, then large angle ACD is called an exterior angle of large Delta ABC at C and is denoted by ext. large Delta ACD.

Theorem:  If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

Given:  A triangle ABC. D is a point on BC produced, forming an exterior angle large angle 4.

To Prove: large angle 4=angle 1+angle 2

                i.e., large angle ACD=angle CAB+angle CBA.

Proof:  In triangle ABC, we have

             large angle 1+angle 2+angle 3=180^{circ}          ...........(i)

Also,    large angle 3+angle 4=180^{circ}              [large because angle 3 and large angle 4 forms a linear pair]  ...(ii)

Find (i) and (ii), we have

           large angle 1+angle 2+angle 3=angle 3+angle 4

large Rightarrow;;;angle 1+angle 2=angle 4

Hence,  large angle 4=angle 1+angle 2 

    i.e. large angle ACD=angle CAB+angle CBA

Illustration: in the figure PQ || RS. Find angle QOR.

Solution: Extend PQ to N

angle PNR = angle SRN= 120^0

    [ Alternate interior angles are equal PN || RS and RO is the transversal ]

angle PNR + angle QNO= 180^0  [ Linear Pair]

120^0 + angle QNO= 180^0

Rightarrow angle QNO= 180^0-120^0=60^0

Now, In bigtriangleup NOQ, we have

angle PQO= angle QNO + angle NOQ              [ Exterior angle Property]

100^0= angle 60^0 + angle NOQ

angle NOQ= 100^0-60^0= 40^0

 


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