By giving a counter-example, show that the following statement is false :
Question:

By giving a counter-example, show that the following statement is false :

p : If all the sides of a triangle are equal, then the triangle is obtuse angled.

 

Solution:

By the properties of triangles, if all the sides of a triangle are equal, then the each of the angle of the triangle will also be equal.

By the question,

All sides of the triangle are equal.

$\therefore$ All angles of the triangle are also equal.

Let each angle of the equilateral triangle be $x^{\circ} .$ We know that the sum of all angles of a triangle is $360^{\circ}$.

$x^{\circ}+x^{\circ}+x^{\circ}=360^{\circ}$

$\rightarrow 3 x^{\circ}=360^{\circ}$

$\rightarrow x^{\circ}=(360 \div 3)^{\circ}$

$\therefore \mathrm{x}^{\circ}=60^{\circ}$

Thus, all angles of the triangle measure $60^{\circ}$ which is an acute angle (lying between $0^{\circ}$ and $90^{\circ}$.)

Obtuse angles are those which lie between $90^{\circ}$ and $180^{\circ} .$

Thus, when all sides are equal in a triangle, its angles measure $60^{\circ}$ each. This implies that all angles are acute angles and not obtuse angles.

Thus, the statement p is false.

 

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