Question

Find the value of x in the following triangle.

Answer 1

Hint: In question it is given that we have a triangle named $\vartriangle ABC$ with angle measure $\angle ABC={{24}^{{}^\circ }}$ , $\angle CAB={{32}^{{}^\circ }}$ and $\angle BCA={{x}^{{}^\circ }}$ and we have to calculate measure of $\angle BCA$ that is value of x which we can do easily with property of triangle known as interior sum of triangle is equals to ${{180}^{{}^\circ }}$ .

Complete step-by-step answer:
ow, from figure we can see that in triangle $\vartriangle ABC$ , $\angle ABC={{24}^{{}^\circ }}$ , $\angle CAB={{32}^{{}^\circ }}$ and $\angle BCA={{x}^{{}^\circ }}$
Also , $\angle ABC\ne \angle CAB$ so triangle $\vartriangle ABC$ is a scalene triangle because $\angle ABC\ne \angle CAB$ which conclude sides $BC\ne CA\ne AB$.
There is a triangle property of triangles which is known as interior angle sum property. interior angle sum property states that if we have a triangle say $\vartriangle ABC$, does not matter it is equilateral triangle, isosceles triangle or scalene triangle, the sum of all three interior angles namely $\angle ABC$ , $\angle CAB$ and $\angle BCA$ is always equals to ${{180}^{{}^\circ }}$ that is $\angle ABC+\angle CAB+\angle BCA={{180}^{{}^\circ }}$
Now, for the figure given in question we have a triangle named $\vartriangle ABC$ with angle measure $\angle ABC={{24}^{{}^\circ }}$ , $\angle CAB={{32}^{{}^\circ }}$ and $\angle BCA={{x}^{{}^\circ }}$ and we have to calculate measure of $\angle BCA$ that is value of x .
So, using the property of interior angle sum property in triangle $\vartriangle ABC$ , we get
$\angle ABC+\angle CAB+\angle BCA={{180}^{{}^\circ }}$……. ( i )
Substituting values of which are given in figure as $\angle ABC={{24}^{{}^\circ }}$ , $\angle CAB={{32}^{{}^\circ }}$ and $\angle BCA={{x}^{{}^\circ }}$ in equation ( i ), we get
${{32}^{{}^\circ }}+{{24}^{{}^\circ }}+{{x}^{{}^\circ }}={{180}^{{}^\circ }}$ ,
On simplifying , we get
${{56}^{{}^\circ }}+{{x}^{{}^\circ }}={{180}^{{}^\circ }}$ ,
Shifting ${{56}^{{}^\circ }}$ from left hand side to right hand side , we get
${{x}^{{}^\circ }}={{180}^{{}^\circ }}-{{56}^{{}^\circ }}$
Subtracting ${{56}^{{}^\circ }}$ from \[{{180}^{{}^\circ }}\] , we get
${{x}^{{}^\circ }}={{124}^{{}^\circ }}$
So, the measure of angle $\angle BCA={{x}^{{}^\circ }}$ is equals to ${{x}^{{}^\circ }}={{124}^{{}^\circ }}$ by interior angle sum property .


Note: While solving the numerical based on angles of triangle, or any other polygon, one must know the sum of interior angles or that polygon. In the calculation part, we should be careful that there must not be any calculation error as it may change the answer .