Question

In the given figure, the value of x is

(a). ${{125}^{\circ }}$
(b). ${{120}^{\circ }}$
(c). ${{145}^{\circ }}$
(d). ${{135}^{\circ }}$

Answer 1

Hint: Angles formed by a chord in the circle in the same segment are equal to each other. Use the $\Delta BPC$ to get the value of x. The sum of interior angles of a triangle is ${{180}^{\circ }}$. Use the above mentioned details and information to get the value of x.

Complete step-by-step answer:

We know the angles formed by the chord of a circle in the same segment will equal to each other. Now, we can observe the given diagram in the problem and get that the angle formed by the chord in the same segment i.e. $\angle BAD,\angle BCD$ are equal to each other.

Hence, we get
$\angle BCD=\angle BAD$ ………. (i)
As, we are given the values of $\angle DPB,\angle DAB,\angle CBP$ as ${{25}^{\circ }},{{30}^{\circ }},{{x}^{\circ }}$in the given diagram. So we can get value of $\angle BCD$ as
$\angle BCD={{30}^{\circ }},\angle BCP={{30}^{\circ }}$ ………….. (ii)
Now, we know the sum of all the angles of any triangles is ${{180}^{\circ }}$. So, we can write the equation in $\Delta CBP$ as
$\angle BPC+\angle CBP+\angle PCB={{180}^{\circ }}$
We know the values of above three angles from the problem and from the equation (ii). So, we can get above equation as
$\begin{align}
  & {{25}^{\circ }}+x+{{30}^{\circ }}={{180}^{\circ }} \\
 & x+{{55}^{\circ }}={{180}^{\circ }} \\
\end{align}$
Now, subtract ${{55}^{\circ }}$from both sides of the above equation, we get
x + 55 – 55 = 180 – 55
x = ${{125}^{\circ }}$
Hence, value of $x\to {{125}^{\circ }}$
So, option (a) is the correct answer.

Note: Another approach for the question would be we can calculate angle $\angle ADP$ in the $\Delta ADP$ and hence, get $\angle CDA$ using linear pair property of a line. Now get $\angle ABC,\angle AQB$ and hence $\angle CQD$. Now, use the angle sum property of a triangle in $\Delta CQD\to \angle BCD$and hence, get the value of x. so, without using property, we will be able to get the answer and this approach will involve a lot of angles.
Using the property that angles in the same segment are equal, is the key point of this question.