Question

In the given figure , AC = BC . Find the value of x and y ?

Answer 1

Hint:
we are given that the sides AC and BC are equal. Using the property that in an isosceles triangle the angles made by the equal sides are equal, we get $\angle BAC = \angle ABC$ and using the property that the angle formed by a straight line is ${180^ \circ }$ we get the value of $\angle ABC$ using which we get y and then using the property sum of the angles in a triangle is ${180^ \circ }$ we get the value of x.

Complete step by step solution:
In the given triangle ABC

We are given that AC = BC
This means that it is an isosceles triangle
And in an isosceles triangle the angles made by the equal sides are equal
$ \Rightarrow \angle BAC = \angle ABC$
We are given that the $\angle BAO = {125^ \circ }$
We know that the angle formed by a straight line is ${180^ \circ }$
$
   \Rightarrow \angle BAC + \angle BAO = {180^ \circ } \\
   \Rightarrow \angle BAC + {125^ \circ } = {180^ \circ } \\
   \Rightarrow \angle BAC = {180^ \circ } - {125^ \circ } \\
   \Rightarrow \angle BAC = {55^ \circ } \\
 $
Now since we know that $\angle BAC = \angle ABC$
We get $\angle ABC = {55^ \circ }$
Once again using the property that the angle formed by a straight line is ${180^ \circ }$
$
   \Rightarrow \angle ABC + y = {180^ \circ } \\
   \Rightarrow {55^ \circ } + y = {180^ \circ } \\
   \Rightarrow y = {180^ \circ } - {55^ \circ } \\
   \Rightarrow y = {125^ \circ } \\
 $
Now we know that the sum of the angles in a triangle is ${180^ \circ }$
$
   \Rightarrow \angle BAC + \angle ABC + \angle BCA = {180^ \circ } \\
   \Rightarrow {55^ \circ } + {55^ \circ } + x = {180^ \circ } \\
   \Rightarrow {110^ \circ } + x = {180^ \circ } \\
   \Rightarrow x = {180^ \circ } - {110^ \circ } \\
   \Rightarrow x = {70^ \circ } \\
 $

Hence we get the values of x and y to be ${70^ \circ }{\text{ and }}{125^ \circ }$

Note:
1) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
2) The side opposite to the largest angle is the longest side of the triangle and the side opposite to the smallest angle is the shortest side of the triangle.
3) An exterior angle of a triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.