Question

If the sides of the triangle are produced in order,prove that the sum of the Exterior angles so formed is equal to T four right angles. R Q​

Answer 1

Step-by-step explanation:

Given: △ABC in which AB,BC and CA are produced to points D,E and F.

To prove: ∠DCA+∠FAE+∠FBD=180

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Proof :

From the figure we know that

∠DCA=∠A+∠B.(1)

∠FAE=∠B+∠C.(2)

∠FBD=∠A+∠C.(3)

By adding equation (1),(2) and (3) we get

∠DCA+∠FAE+∠FBD=∠A+∠B+∠B+∠C+∠A+∠C

So we get

∠DCA+∠FAE+∠FBD=2∠A+2∠B+2∠C

Now by taking out 2 as common

∠DCA+∠FAE+∠FBD=2(∠A+∠B+∠C)

We know that the sum of all the angles in a triangle is 180

∘

.

So we get

∠DCA+∠FAE+∠FBD=2(180

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)

∠DCA+∠FAE+∠FBD=360

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Therefore, it is proved.

Answer 2

Step-by-step explanation:

Let a, b, c be the measure of interior angles and x , y, z be the measures of exterior angles.

Therefore,

a + b + c = 180°.

x = b + c,

y = a + c,

z = a + b

Hence,

x + y + z = 2a + 2b + 2c

x + y + z = 2(a + b + c)...(2)

From (1) and (2)

x + y + z = 2 * 180°

x + y + z = 360°

x + y + z = 4 x 90°

x + y + z = 4 right angles

Thus the sum of the exterior angles of a triangle is equal to four right angles