Question

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Answer 1

Question:-

If in ∆ABC , 2∠A = 3∠B = 6∠C. Find the ∠A , ∠B and ∠C of ∆ABC.

Answer:-

Given:

In a ∆ABC ,

2∠A = 3∠B = 6∠C

Let, 2∠A = 3∠B = 6∠C = k

• 2∠A = k ⟹ ∠A = k/2

• 3∠B = k ⟹ ∠B = k/3

• 6∠C = k ⟹ ∠C = k/6

We know that,

Sum of three angles of a triangle = 180°

⟹ ∠A + ∠B + ∠C = 180°

⟹ k/2 + k/3 + k/6 = 180°

⟹ (3k + 2k + k) / 6 = 180°

⟹ 6k/6 = 180°

⟹ k = 180°

∴

• ∠A = k/2 = 180°/2 = 90°

• ∠B = k/3 = 180°/3 = 60°

• ∠C = k/6 = 180°/6 = 30°

Answer 2

$\begin{gathered}\begin{gathered}\bf \: Given\:-\begin{cases} &\sf{In \: \triangle \: ABC} \\ &\sf{2 \angle \: A = 3\angle \: B = 6\angle \: C} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\:find \: the \: values \: of \: \:- \begin{cases} &\sf{\angle \: A}  \\ &\sf{\angle \: B}  \\ &\sf{\angle \: C}  \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

$\begin{gathered}\begin{gathered}\bf \:Let- \begin{cases} &\sf{2 \angle \: A = 3\angle \: B = 6\angle \: C = k}  \end{cases}\end{gathered}\end{gathered}$

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: A\: = \tt \: \dfrac{k}{2} }}$

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: B \: = \tt \: \dfrac{k}{3} }}$

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: C \: = \tt \: \dfrac{k}{6} }}$

Now,

We know, that

• Sum of angles of a triangle is supplementary.

So,

$\longrightarrow{\blue{ \bf \: In \: \triangle \: ABC}}$

$\rm :\implies\:\angle \: A \: + \angle \: B \: + \angle \: C = 180 \degree$

$\rm :\implies\:\dfrac{k}{2} + \dfrac{k}{3} + \dfrac{k}{6} = 180 \degree$

$\rm :\implies\:\dfrac{3k + 2k + k}{6} = 180 \degree$

$\rm :\implies\:\dfrac{6k}{6} = 180 \degree$

$\rm :\implies\:k \: = \: 180 \degree$

Hence,

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: A\: = \tt \: \dfrac{k}{2} = \dfrac{180}{2} = 90 \degree }}$

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: B\: = \tt \: \dfrac{k}{3} = \dfrac{180}{3} = 60 \degree }}$

$\rm :\implies\: \boxed{ \pink{ \bf \: \angle \: C\: = \tt \: \dfrac{k}{6} = \dfrac{180}{6} = 30 \degree }}$

More information :-

Properties of a triangle

• A triangle has three sides, three angles, and three vertices.

• The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

• The side opposite to the largest angle of a triangle is the largest side.

• Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

• Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

• Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.