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If the side BC of $Δ A B C$ is produced on both sides, then the difference between the sum of the exterior angles so formed and $∠ A$ is ____ $°$ .

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Detailed Solution

If the side BC of

Δ A B C

is produced on both sides, then the difference between the sum of the exterior angles so formed and

∠ A

is 180

°

.
Given the side BC of

Δ A B C

is produced on both sides.
Question Image

Using angle sum property in ABC,

∠ A C B + ∠ B A C + ∠ C B A = 180 °

Exterior angle property states that if a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Using Exterior Angle property,

E x t . ∠ A C D = ∠ A + ∠ C B A … … (1) E x t . ∠ A B E = ∠ A + ∠ A C B … … (2)

Adding both equations,

⇒ E x t . ∠ A C D + E x t . ∠ A B E = ∠ A + ∠ C B A + ∠ A + ∠ A C B ⇒ E x t ⋅ ∠ A C D + E x t . ∠ A B E = 2 ∠ A + ∠ C B A + ∠ A C B … … (3)

Using angle sum property in triangle ABC,

∠ A + ∠ C B A + ∠ A C B = 180 ° … … (4)

Difference is,

⇒ (E x t . ∠ A C D + E x t . ∠ A B E) − ∠ A = 2 ∠ A + ∠ C B A + ∠ A C B − ∠ A ⇒ (E x t . ∠ A C D + E x t . ∠ A B E) − ∠ A = ∠ A + ∠ C B A + ∠ A C B … … (5)

⇒ (Ext . ∠ ACD + Ext . ∠ ABE) − ∠ A = 180 °

Hence, the difference is 180°.

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