In a ∆ABC, ∠ C = 3 ∠ B = 2 (∠A + ∠ B). Find the three angle
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Answer:
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C O N C E P T :
★ In this question first let us suppose that 2∠A=3∠B=6∠C = x . Now try to find out the values of ∠ A , ∠ B and ∠C in the terms of x After that we know that the sum of the interior angles of the triangle is 180∘ . By using this property we will find out the value of x and the remaining angles.
★ Whenever we have found some value of angle, always consider that it is equal to x . Now try to find out some relation between them and use the properties of triangles to proceed further . As in this the angle A is 90∘ hence it is a right angle triangle .
Equilateral Triangle : In which all the sides are equal in length . In this triangle all the angles are 60∘.
S O L U T I O N :
➽ ∠C = 3∠B = 2(∠A + ∠B)
➽ 3∠B = 2(∠A + ∠B)
➽ 3∠B = 2∠A + 2∠B
➽ ∠B = 2∠A
➽ 2 ∠A − ∠B = 0 … (i)
We know that the sum of the measures of all angles of a triangle is 180°.
Therefore,
➽ ∠A + ∠B + ∠C = 180°
➽∠A + ∠B + 3 ∠B = 180°
➽∠A + 4 ∠B = 180° … (ii)
➽Multiplying equation (i) by 4, we obtain
8 ∠A − 4 ∠B = 0 … (iii)
➽ Adding equations (ii) and (iii), we obtain
9 ∠A = 180°
➽ ∠A = 20°
From equation (ii), we obtain
➽ 20° + 4 ∠B = 180°
➽ 4 ∠B = 160°
➽ ∠B = 40°
➽ ∠C = 3 ∠B
➽ = 3 × 40° = 120°