Question

Find the value of x in the adjoining figure:

Answer 1

Hint: We will use the property of isosceles triangles and the Angle Sum Property as well to calculate the value of the unknown angle x.

Complete step-by-step answer:
From the given figure, we can deduce that the sides AB and AC of the triangle are equal. We can say that the given triangle ABC is an isosceles triangle.
Property of isosceles triangles: The angles opposite to the equal sides are equal.
From the property we can say that $\angle $ABC = $\angle $ACB.
We are given that $\angle $ACD = 125$^ \circ $and $\angle $BAC = x.
We can see from the figure that $\angle $BCD = 180$^ \circ $
$ \Rightarrow $$\angle $BCD = $\angle $ACB + $\angle $ACD
$ \Rightarrow $180$^ \circ $= $\angle $ACB + 125$^ \circ $
$ \Rightarrow $$\angle $ACB = 180$^ \circ $- 125$^ \circ $= 55$^ \circ $
$ \Rightarrow $$\angle $ACB = $\angle $ABC = 55$^ \circ $
By the Angle Sum Property, we have the statement: the sum of all sides of a triangle must be equal to 180$^ \circ $.
$\therefore $ $\angle $ABC + $\angle $ACB + $\angle $BAC = 180$^ \circ $
$ \Rightarrow $55$^ \circ $+ 55$^ \circ $+ x = 180$^ \circ $
$ \Rightarrow $x = 180$^ \circ $- 110$^ \circ $
$ \Rightarrow $x = 70$^ \circ $
Therefore, the value of the unknown angle in the given isosceles triangle x is found to be 70$^ \circ $.

Note: In such problems, we first analyse the question and then we proceed since we know an unknown angle to be calculated with help of an exterior angle of the triangle. As a triangle is isosceles so angles opposite to equal sides are of equal measure. Once we get the angle value we will apply angle sum property.