Question

The angle of a triangle are AP.The smallest angle is 30degree.show that it is a right angle triangle

Answer 1

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• The angle of a triangle are AP.The smallest angle is 30degree.show that it is a right angle triangle.

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Let D is the difference forming an AP.

• If 1st angle is 30,

then,

• 2nd angle will be 30+D,

Similarly,

• 3rd angle will be 30+2D.

Adding these 3 angles gives 180 which is addition of all 3 angles of a triangle.

So ,

$\sf \implies \: 30+30+D+30+2D =180 \\\sf \implies \: 90+3D= 180 \\ \sf \implies \: 3D= 180-90 \\ \sf \implies \: 3D=90 \\ \sf \implies \: D= \cancel\frac{90}{3} \\ \sf \implies \: D= 30$

• $\sf \: 2nd \: angle \: will \: be \: 30+30= {60}^{ \circ}$
• $\sf \:3rd\:Angle=30+2\times30={90}^{ \circ}$

As we know, 90⁰ represents the right angle so it is proved that the triangle is a right angled triangle as the 3rd angle is 90⁰.

Answer 2

Given :

• The angles of a triangle are in AP.
• The smallest triangle is 30°

To Prove :

It is a right angle triangle.

Solution :

Let ABC be a traingle in which angles are in AP.

Thus ,∠A = x ,∠B=x+a and ∠C=x+2a

where , a is the common difference between angles of triangle.

According to the question:

Smallest angle = 30°

Then ,

• ∠A= 30°
• ∠B = 30°+ x
• ∠C= 30° +2x

We know that :

sum of all angles of triangle = 180°

∠A+∠B+∠C=180°

30° +30°+x +30°+2x = 180°

90° +3x = 180°

3x = 90°

x = 30°

Therefore ,

∠A= 30°

∠B = 30°+ x = 30+30= 60°

∠C= 30° +2× 30= 90°

Hence ,Proved ∆ABC is a right angled triangle,

right angle at C .