Question

the smallest angle of a triangle is two-thirds the size of the middle angle, and the middle angle is three-sevenths of the largest angle. Find all three angle measures​

Answer 1

Let the largest angle = x

middle angle = 3x/7

smallest angle = 2/3 × 3x/7 = 2x/7

sum of the angles of a triangle = 180⁰

2x/7 + 3x/7 + x = 180⁰

2x + 3x + 7x = 180

7

12x = 180

7

12x = 180 × 7

x = 180 × 7

12

x = 105⁰

3x/7 = 3 x 105 = 45⁰

7

2x/7 = 2 × 105 = 30⁰

7

The smallest angle = 30⁰

The middle angle = 45⁰

The largest angle = 105⁰

Answer 2

$\LARGE\textbf{\underline{\underline{Solution :-}}}$

$\large\boxed{\texttt\textcolor{green}{↣ Let the largest angle be ' 7x '. }}\\\\\\\large\texttt{↣ So middle angle will be = \cfrac{3}{7} × 7x }\\\\\large\texttt{= \cfrac{3}{\cancel{7}} × \cancel {7x} }\\\\\large\boxed{\therefore\texttt\textcolor{green}{Middle angel will be = 3x }}\\\\\\\large\texttt{↣ So the small angle will be = \cfrac{2}{3} × 3x }\\\\\large\texttt{= \cfrac{2}{\cancel{3}} × \cancel{3x} }\\\\\large\boxed{\therefore\texttt\textcolor{green}{Small angle will be = 2x} }$

• As we know that the sum of all angles of the triangle is 180° .

• So,

$\large:\: \bigstar\textsf\textcolor{orange}{\: \: \: 7x° + 3x° + 2x° = 180° }\\\\\\\large: \: \Longrightarrow\textsf{12x° = 180°}\\\\\\\large: \: \Longrightarrow\textsf{x = \cfrac{180}{12} }\\\\\\\large: \: \Longrightarrow\textsf{x = \cancel\cfrac{180}{12} }\\\\\\\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{x° = 15°}}}$

$\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{voilet}{Small angle = 2 × 15 = 30°}}}\\\\\\\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{purple}{Middle angel= 3 × 15 = 45° }}}\\\\\\\large : \: \Longrightarrow\underline{\boxed{\textsf\textcolor{magenta}{Largest angle = 7 × 15 = 105°}}}$