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Question

If the sides of a triangle are produced in order, prove that the sum of the exterior angles so formed is equal to four right angles.


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Solution

Prove the required statement:

Let a triangle ABC in which the sides areAB,BC,CA produced F,D and E.

We have to prove DCA+FAE+FBD=360°.

By using the theorem, an exterior angle of a triangle is equal to the sum of two remote interior angles.

From the Figure, we can write,

A+B=DCA . . . . . . 1

B+C=FAE . . . . . . 2

A+C=FBD . . . . . . 3

Adding all three equations, we get

A+B+B+C+A+C=DCA+FAE+FBD

2A+2B+2C=DCA+FAE+FBD

2(A+B+C)=DCA+FAE+FBD . . . . . . . 4

Since we know that sum of a triangle is 180°

So, A+B+C=180°

By putting the value in equation 4, we get

2(180°)=DCA+FAE+FBD

360°=DCA+FAE+FBD

90°×4=DCA+FAE+FBD

4rightangles=DCA+FAE+FBD

Hence, proved that the sum of the exterior angles so formed is equal to four right angles.


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