Solveeit Logo

Question

Question: In the given figure A, B, C and D are concyclic points. The value of x is ![](https://www.vedantu....

In the given figure A, B, C and D are concyclic points. The value of x is

[a] 5050{}^\circ
[b] 6060{}^\circ
[c] 7070{}^\circ
[d] 9090{}^\circ

Explanation

Solution

Hint: Find the measure of CBA\angle CBA using the fact that the angles CBA and CBF form a linear pair. Using the fact that the sum of opposite angles of a cyclic quadrilateral, find the measure of angle CDA. Finally, using the fact that the angles CDA and CDE from a linear pair find the measure of angle ECD. Hence find the value of x. Alternatively, use the fact that the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle and hence find the measure of x.

Complete step-by-step answer:

Given: ABCD is a cyclic quadrilateral. Side AB is extended to point F, and AD is extended to point E. The measure of angle FBC is 130130{}^\circ , and the measure of angle ECD is x.
To determine: The value of x.
Since ABF are collinear, we have angles CBA and CBF form a linear pair
Hence, we have
CBF+CBA=180\angle CBF+\angle CBA=180{}^\circ
Substituting the value of CBF\angle CBF, we get
130+CBA=180130{}^\circ +\angle CBA=180{}^\circ
Subtracting 130 on both sides, we get
CBA=50\angle CBA=50{}^\circ
Now, we know that the sum of measures of opposite angles of a cyclic quadrilateral is 180180{}^\circ . Since angles CDA and CBA are opposite angles of the cyclic quadrilateral ABCD, we have
CDA+CBA=180\angle CDA+\angle CBA=180{}^\circ
Substituting the value of CBA,\angle CBA, we get
CDA+50=180\angle CDA+50{}^\circ =180{}^\circ
Subtracting 50 from both sides of the equation, we get
CDA=130\angle CDA=130{}^\circ
Now, since A,D and E are collinear, we have the angels EDC and CDA form a linear pair
Hence, we have
EDC+CDA=180\angle EDC+\angle CDA=180{}^\circ
Substituting the value of EDC\angle EDC and CDA\angle CDA, we get
x+130=180x+130{}^\circ =180{}^\circ
Subtracting 130 on both sides, we get
x=50x=50{}^\circ
Hence the value of x is 5050{}^\circ
Hence option [a] is correct.

Note: Alternative solution:
We know that the exterior angle of the cyclic quadrilateral is equal to the interior opposite angle.
Since EDC\angle EDC is an exterior angle of the cyclic quadrilateral ABCD and CBA\angle CBA is the corresponding interior opposite angle, we have
x=CBA=50x=\angle CBA=50{}^\circ
Hence, the value of x is 5050{}^\circ