Question

An exterior angle of a triangle is 105° and its its interior opposite angles are in the ratio of 3:4. Find the angles of the triangle

Answer 1

Answer:

45,60,75

Step-by-step explanation:

Answer 2

$\huge\mathbb{\underline{\underline{SOLUTION:-}}}$

$\bold{Given}\begin{cases}\sf{Exterior\: angle\:of\: triangle}\\ \sf{\implies 105{}^{\circ}}\end{cases}$

• Exterior angle of a triangle is equal to sum of the two opposite interior angles of triangle

$\sf\underline{\underline{\red{According\: to\: Question:-}}}$

$\sf Two\:interior\: angles\:in\:Ratio\\ \\ \sf\implies 3:4$

• let the interior angles be x

• $\sf so,\: it's \:3x {}^{\circ}\:and \:4x {}^{\circ}$

$\sf \implies 3x{}^{\circ}+4x{}^{\circ}=105{}^{\circ}\\ \\ \sf\implies 7x=105{}^{\circ}\\ \\ \sf\implies x=\cancel{\frac{105{}^{\circ}}{7}}=15{}^{\circ}\\ \\ \sf\longrightarrow The\:value\:of\:x=15{}^{\circ}$

• So interior angles are:-

• $\sf 3\times 15{}^{\circ}=45{}^{\circ}$
• $\sf 4\times 15{}^{\circ}=60{}^{\circ}$

$\tt The\:two\: angles\:of\: triangle=45{}^{\circ}\:and\:60{}^{\circ}$

• Now we have to find the third angle of triangle.

• We know that the sum of angles of triangle is$\sf 180{}^{\circ}$

So ,

• Let the third angle of triangle be a

$\sf 45{}^{\circ}+60{}^{\circ}+a=180{}^{\circ}\\ \\ \sf\implies 105{}^{\circ}+a=180{}^{\circ}\\ \\ \sf\implies a=180{}^{\circ}-105{}^{\circ}\\ \\ \sf\implies a=75{}^{\circ}$

• The value of a is $\sf 75{}^{\circ}$

• So, a is the third angle of triangle

• The third angle of triangle is

$\sf\implies{75 {}^{\circ}}$

$\boxed{\underline{\mathfrak{\purple{Angles\:of \triangle=75{}^{\circ}\:60{}^{\circ}\:45{}^{\circ}}}}}$

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