Question
If the three angles of a triangle are: $${\left(x + 15 \right)^ \circ },$$ $${\left({\frac{{6x}}{5} + 6} \right)^ \circ }$$ and $${\left({\frac{{2x}}{3} + 30} \right)^ \circ }$$ then the triangle is:
Answer: Option B
According to question,
$$ \Rightarrow \left( {x + {{15}^ \circ }} \right) + {\left( {\frac{{6x}}{5} + 6} \right)^ \circ } + $$ $${\left( {\frac{{2x}}{3} + 30} \right)^ \circ } = $$ $${180^ \circ }$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\angle A + \angle B + \angle C = {{180}^ \circ }} \right\}$$
$$ \Rightarrow x + \frac{{6x}}{5} + \frac{{2x}}{3} = $$ $${180^ \circ } - \left(15 + 6 + 30\right)$$
$$\eqalign{
& \Rightarrow \frac{{15x + 18x + 10x}}{{15}} = 180 - 51 \cr
& \Rightarrow 43x = 129 \times 15 \cr
& \Rightarrow x = {45^ \circ } \cr
& \Rightarrow {\text{each}}\,{\text{angle}} \cr
& \Rightarrow {\left( {x + 15} \right)^ \circ } = 45 + 15 = {60^ \circ } \cr
& \Rightarrow {\left( {\frac{{6x}}{5} + 6} \right)^ \circ } = {60^ \circ } \cr
& \Rightarrow {\left( {\frac{{2x}}{3} + 30} \right)^ \circ } = {60^ \circ } \cr} $$
∵ All three angles are equal 60°
⇒ Triangle will be equilateral triangle
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According to question,
$$ \Rightarrow \left( {x + {{15}^ \circ }} \right) + {\left( {\frac{{6x}}{5} + 6} \right)^ \circ } + $$ $${\left( {\frac{{2x}}{3} + 30} \right)^ \circ } = $$ $${180^ \circ }$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\angle A + \angle B + \angle C = {{180}^ \circ }} \right\}$$
$$ \Rightarrow x + \frac{{6x}}{5} + \frac{{2x}}{3} = $$ $${180^ \circ } - \left(15 + 6 + 30\right)$$
$$\eqalign{
& \Rightarrow \frac{{15x + 18x + 10x}}{{15}} = 180 - 51 \cr
& \Rightarrow 43x = 129 \times 15 \cr
& \Rightarrow x = {45^ \circ } \cr
& \Rightarrow {\text{each}}\,{\text{angle}} \cr
& \Rightarrow {\left( {x + 15} \right)^ \circ } = 45 + 15 = {60^ \circ } \cr
& \Rightarrow {\left( {\frac{{6x}}{5} + 6} \right)^ \circ } = {60^ \circ } \cr
& \Rightarrow {\left( {\frac{{2x}}{3} + 30} \right)^ \circ } = {60^ \circ } \cr} $$
∵ All three angles are equal 60°
⇒ Triangle will be equilateral triangle
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