Question

The triangle \[ABC\] is a right-angled triangle in which \[\angle A={{90}^{0}}\] and \[AB=AC\]. Find \[\angle B\] and \[\angle C\].

Answer 1

Hint: To solve the question represent the given information in a triangle diagram, which will help to understand and analyse the sides and angles relation. The question is a conclusion drawn from the properties of isosceles right-angles triangles.

Complete step-by-step answer:
\[ABC\] is a right-angled triangle at \[\angle A={{90}^{0}}\].
The two adjacent sides of the triangle \[ABC\] are equal.
\[AB=AC\]

We know that angles opposite to the equal sides of a triangle are equal.
The opposite angle to side \[AB\] is\[\angle C\]and the opposite angle to side \[AC\] is \[\angle B\].
\[\Rightarrow \angle B=\angle C\]
We know that the sum of all angles of a triangle is equal to \[{{180}^{0}}\].
\[\Rightarrow \angle A+\angle B+\angle C={{180}^{0}}\]
By substituting the given and evaluated values of the angles of the triangle \[ABC\] we get,
\[{{90}^{0}}+\angle B+\angle B={{180}^{0}}\]
\[2\angle B={{180}^{0}}-{{90}^{0}}\]
\[2\angle B={{90}^{0}}\]
\[\angle B=\dfrac{{{90}^{0}}}{2}={{45}^{0}}\]
\[\therefore \angle B=\angle C={{45}^{0}}\]
Thus, the angles \[\angle B\] and \[\angle C\] of right-angled triangle \[ABC\] are equal \[{{45}^{0}}\].

Note: To solve the question we need to analyse the given information to draw the conclusion that the two sides of the triangle are equal implies that the unknown two angles are equal. The conclusion is drawn by knowing the basic theorems of triangles which states angles opposite to the equal sides of a triangle are equal. For the further solving apply the formula of the sum of all the angles of the triangle. While solving, remember to solve step wise to get the calculations right. The other possibility of mistake can be a calculation mistake which can be self-checked as the sum of all angles of the triangle is equal to \[{{180}^{0}}\].