Question

If the sides of a triangle are produced in order prove that the sum of the exterior angle so formed is equal to four right angles.
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Answer 1

Question:—

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

Answer:—

• → 90°

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°_________________(1)

Now by remote exterior angle theorem,

We have:

→ 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.

Proved.

$@vikramjeeth$

Answer 2

Answer:

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

To prove: ∠DCA+∠FAE+∠FBD=180∘

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∘)

∠DCA+∠FAE+∠FBD=360∘

Therefore, it is proved.

Step-by-step explanation:

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