Question

If A, B, C and D are concyclic points and $\angle BAC={{45}^{0}}$. Find the value of $\angle BDC$
A.${{45}^{0}}$
B.${{60}^{0}}$
C.${{75}^{0}}$
D.${{90}^{0}}$

Answer 1

Hint: If A, B, C and D are concyclic then these points are passing through a circle. Now, draw a circle passing through these points and it is given that $\angle BAC={{45}^{0}}$ so $\angle BDC$ will be found out by the property that the points on the circle which are lying on the same side of the given chord are subtending equal angles on that chord.

Complete step-by-step answer:
It is given that A, B, C and D are concyclic so we can draw a circle passing through these points.

In the above diagram, A, B, C and D are lying on the circle and $\angle BAC={{45}^{0}}$.
We have to find $\angle BDC$. As you can see that $\angle BDC\And \angle BAC$ lie on the same side of the chord BC so we can use the property that which says that if A, B, C and D are concyclic and points A and D lie on the same side of the chord (or side) BC then angle subtended by the points lying on the same side of the side are equal.
As $\angle BDC\And \angle BAC$ lie on the same side of the chord BC so using the property of concyclic points that we have just discussed we can say that both the angles are equal.
So, $\angle BDC=\angle BAC={{45}^{0}}$.
Hence, the correct option is (a).

Note: If you don’t know the property of concyclic points. In the greed of getting more marks don’t try to consider the $\angle B\And \angle C$ as ${{90}^{0}}$ and then try to get the value of $\angle BDC$. You might think if you assume these angles as ${{90}^{0}}$ you will somehow get the answer. Don’t do such adventures in the exam, you might take the risk, if the question has no negative marking. Moreover it is advisable to know what concyclic points mean.