Question

The angles of atriangle are in ap and the tangent of the smallest angle is 1.find the other angles of the triangles

Answer 1

$\mathcal{\huge{\fbox {\red{AnSwEr:-}}}}$

➡ The following angles of triangle are 60° , 45° & 75°.

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$\mathcal{\huge{\fbox{\purple{Solution:-}}}}$

Let the angles of triangle be,

• x
• x-d
• x+d

We, know that the sum of angles of a triangle is :-

➡ a + b + c = 180°

➡ x + x - d + x + d = 180

➡ 3x - d + d = 180

➡ 3x = 180

➡ x = 180 / 3

➡ x = 60

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Using tan a = 1

Tan 45° =1

Now, tan(x-d) = 1

▶ (x-d) = 45

▶ (60-d) = 45

▶ - d = 45 - 60

▶ -d = -15

▶ d = 15

Therefore, d = 15

• x = 60°

• x-d = 60 - 15 = 45°

• x+d = 60 + 15 = 75°

➡ The other angles of a triangle are, 60° , 45° & 75°.

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$\mathcal{\huge{\fbox{\green{VERICATION:-}}}}$

➡ a + b + c = 180°

➡ 60° + 45° + 75° = 180°

➡ 180° = 180°

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Additional information :-

• A triangle a three sided polygon.

• It is of different types :- Right triangle, isosceles triangle, equilateral triangle, Scalene, Obtuse triangle & acute triangle

• The sum of angles of a triangle is 180°.

• A triangle also plays a basic role in trigonometry.

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Answer 2

Let us consider the angles to be :

$\longrightarrow{\tt{x}}$

$\longrightarrow{\tt{x - d}}$

$\longrightarrow{\tt{x + d}}$

We have taken (x- d) and (x + d) as the tangent of smallest angle is specified .

We know that the sum of angles is always equal to 180° .

So we will write the angles in equation form :

$\longrightarrow{\tt{x + x - d + x + d = 180}}$

Adding up like terms of x :

$\longrightarrow{\tt{3x + d - d = 180}}$

Opposite terms get cancelled :

$\longrightarrow{\tt{3x \cancel{-d + d} = 180}}$

So we get :

$\longrightarrow{\tt{3x = 180}}$

To find the value of x =

$\longrightarrow{\tt{x = \dfrac{180}{3}}}$

So , x =

$\longrightarrow{\tt{x = \cancel{\dfrac{180}{3}}}}$$\longrightarrow{\tt{x = 60°}}$

Now to find the value of d :

tan(x - d) = 1

Putting up the value of x =

(60 - d ) = 45

Transposing 60 :-

-d = 45 - 60

Simplify :

-d = -15

So d :15

As we have found the value of x we can follow the same to make (x - d) and (x + d)

$\longrightarrow{\tt{x = 60°}}$

$\longrightarrow{\tt{x - 15 = 60° - 15 = 45°}}$

$\longrightarrow{\tt{x + 15 = 60° + 15 = 75°}}$