Question

3. If the three angles of a triangle are (x +15degree )(6x/5+6degree)(2x/3+30degree) prove that the triangle is an
equilateral triangle.​

Answer 1

Given :-

• The three angles of a triangle are ( x + 15° ) , ( 6x/5 + 6° ) & ( 2x/3 + 30° )

To Prove :-

• The triangle is an equilateral triangle

Proof :-

• The three angles of a triangle are ( x + 15° ) , ( 6x/5 + 6° ) & ( 2x/3 + 30° )

$\qquad$ ☀ We know that in a triangle , sum of all angles is equal to 180°. In case of equilateral triangle the measures of 3 angles are equal because the all 3 sides are equal and the measurement of each angle will be 60°.

$\qquad$ ☀ In the given question, x is an unknown value . So, if we are able to find the value of x then we can easily prove that this triangle is an equilateral triangle .

$\qquad\small\underline{\pmb{\sf \:According \: to \: the \: question :-}}$

$\pink{\qquad\leadsto\quad \pmb {\mathfrak{x + 15° + \dfrac{6x}{ 5 }+ 6° + \dfrac{2x}{3 }+ 30° = 180° }}}$

$\qquad\leadsto\quad \pmb {\mathfrak{x + \dfrac{6x}{ 5 }+ \dfrac{2x}{3 } + 15° + 6° + 30° = 180° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{x + \dfrac{6x}{ 5 } + \dfrac{2x}{3 } + 51° = 180° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{x + \dfrac{6x}{ 5 }+ \dfrac{2x}{3 }= 180° - 51° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{x + \dfrac{6x}{ 5 } + \dfrac{2x}{3 } = 129° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{15x + 18x + \dfrac{10x}{15} = 129°}}$

$\qquad\leadsto\quad \pmb {\mathfrak{\dfrac{43x}{15 }= 129° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{43x = 129° \times 15 }}$

$\qquad\leadsto\quad \pmb {\mathfrak{43x = 1935° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{x =\cancel{ \dfrac{1935°}{43 }}}}$

$\pink{\qquad\leadsto\quad \pmb {\mathfrak{x = 45°}}}$

• Value of x = 45°

Now, let's find the measurement of each angle ; -

$\underline{\rm{\sf 1st\:Angle:-}}$

$\qquad\leadsto\quad \pmb {\mathfrak{ x + 15° = 45° + 15°= 60°}}$

$\underline{\rm{\sf 2nd\:Angle:-}}$

$\qquad\leadsto\quad \pmb {\mathfrak{ \dfrac{6x}{ 5 }+ 6° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{6 \times 9° + 6° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ 54° + 6° = 60°}}$

$\underline{\rm{\sf 3rd\:Angle:-}}$

$\qquad\leadsto\quad \pmb {\mathfrak{ \dfrac{2x}{3} + 30° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ 2 \times 15° + 30° }}$

$\qquad\leadsto\quad \pmb {\mathfrak{ 30°+30° = 60°}}$

Hence,The measurements of the three angles are ; 60° , 60° & 60°.

• Hence, Proved!!

⠀⠀⠀⠀¯‎¯‎‎‎¯‎¯¯‎¯‎‎‎¯‎‎‎¯‎¯‎¯‎‎¯‎‎‎¯‎¯‎‎‎¯‎‎‎¯‎¯‎¯‎‎‎¯‎‎¯‎‎‎¯‎‎¯‎¯‎‎‎¯‎‎‎¯‎¯‎¯‎‎¯‎¯‎‎¯‎¯¯‎¯‎¯‎‎‎¯‎¯‎‎‎¯‎‎‎¯‎‎‎¯‎‎‎¯‎