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State the following statement is True or False.In an isosceles triangle, any two angles of a triangle are the same.A. TrueB. False 

Last updated date: 18th Sep 2024
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Hint: We recall the definition of the polygon, triangles, side, and the internal angles of the triangle. We use the theorem that the angles opposite to sides of equal length have equal measures and in an isosceles triangle, only two sides are of equal length which means only two angles will have an equal measure, not ANY two angles.

Complete step-by-step solution
The closed curve which is formed only by joining a finite number of line segments is called a polygon and the point of intersection of line segments is called the vertex. Any line segment which joins to form the polygon is called edge or side. The angles subtended by any two sides is called the angle of the polygon. 
A triangle is a polygon with three sides and three vertices otherwise known as 3-gon. If we denote the vertices of any triangle as A, B, C then its sides are denoted as $\overline{AB},\overline{BC},\overline{AC}$ and the length of the sides are denoted as $AB, BC, AC$. If the lengths of the triangle are not equal to each other which means $AB\ne BC\ne AC$ then it is called a scalene triangle.

The angles of the triangle ABC are $\angle ABC,\angle ACB,\angle BAC$. The angles opposite to sides of length $AB, AC, BC$ have the measure denoted as $m\angle ACB,m\angle ABC,m\angle BCA$ respectively.
If the lengths of any two sides are equal we call the triangle isosceles triangle which means one of $AB=BC, BC=AC, AB=AC$ in triangle ABC is true. We draw the figure of an isosceles triangle with $AB=AC$.

We know that the angels opposite to sides of equal length have equal measures which means if $AB=AC$ then $m\angle ACB=m\angle ABC$. In the above triangle, we have $m\angle ACB=m\angle ABC$ but we do not have $m\angle ACB=m\angle BAC$ or $m\angle ABC=m\angle BAC$. So if we choose any two angles they may not be equal which means the statement ‘In an isosceles triangle any two angles of a triangle are the same’ is false. 

Note: We note that If lengths of all sides of a triangle are equal we call the triangle equilateral and in an equilateral triangle any two angles are the same and with measure${{60}^{\circ }}$. The angles formed at the interior of the triangle (or polygon) are also called the interior angle or internal angle. We also note that the vertex common to equal sides in an isosceles triangle is also called the apex.