Question

$\huge \mathfrak \fcolorbox{purple}{lavender}{QUESTION?¿}$

If each a angle of a triangle is less than the sum of other two, show that the triangle is acute angled.

$\huge \tt\fcolorbox{purple}{lavender}{NOTE:-}$

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Answer 1

$\sf \colorbox{lightgreen} {Answer:–}$

To Prove:. if each angle of triangle is less than the sum of the other two show that the triangle is acute angle.

$\textbf{Prove: } \text{Let the angle of Triangle}$

$\text{are A, B and C}$

$\textbf{According to Question}$

$< A < < B + < C→(1)$

$< B < < A + < C→(2)$

$< C < < B + < A→(3)$

$\textbf{Adding < A of the equation(1)}$

$∴2 < A \: < \: < A + < B + < C$

$⇒2 < A < 180°$

$⇒ < A < 90°$

$\textbf{Adding < B of the equation(2)}$

$∴ < B < 90°$

$\textbf{Adding < C of the equation(3)}$

$∴ < C < 90°$

$\fbox{Hence the angle is Acute angle}$

$\\ \\ \\ \\ \\ \sf \colorbox{gold} {\red(ANSWER ᵇʸ ⁿᵃʷᵃᵇ⁰⁰⁰⁸}$

Answer 2

$\Large\bf{Answer:-}$

we know,

• That the sum of all angles of a triangle is 180°

Now,

$\small\mathfrak\pink{According\:to\:the\:question:-}$

• Each angle of the triangle is less than the sum of the other two.

That is

• A < B + C-----------(1)
• B < C + -----------.(2)
• C < A + B-------.(3)

Now,

A + B + C = 180° [Since, the sum of all angles of a triangle is 180°]

= B + C = 180° - A

Now,

By putting the value in equation (1) we obtain,

$\small\sf \implies{A<180°-A}$

$\small\sf\implies{2A<180°}$

$\small\sf\implies{A< 90° }$

For the second case we get,

A + B + c = 180°

c + A = 180°-B

By putting the value in equation (2) we obtain,

$\small\sf\implies{B < C + A }$

$\small\sf\implies{B < 180° -B }$

$\small\sf\implies{2B<180°}$

$\small\sf\implies{B<90°}$

Now,

In the third case,

A + B + C = 180°

A + B = 180° - C

By, putting the value in equation (3) we obtain,

$\small\sf\implies{C < A + B }$

$\small\sf\implies{C < 180° -C}$

$\small\sf\implies{2C <180°}$

$\small\sf\implies{C<90°}$

Here,

we found that all the angles are acute angle (less than 90°)

Hence,

It is probed that the triangle is an acute angled triangle .

$\small\bf\green{Hope\:uh\:got\:me\::)}$