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] 50∘
[b] 60∘
[c] 70∘
[d] 90∘
Solution
Hint: Find the measure of ∠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 130∘ , 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∘
Substituting the value of ∠CBF, we get
130∘+∠CBA=180∘
Subtracting 130 on both sides, we get
∠CBA=50∘
Now, we know that the sum of measures of opposite angles of a cyclic quadrilateral is 180∘. Since angles CDA and CBA are opposite angles of the cyclic quadrilateral ABCD, we have
∠CDA+∠CBA=180∘
Substituting the value of ∠CBA, we get
∠CDA+50∘=180∘
Subtracting 50 from both sides of the equation, we get
∠CDA=130∘
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∘
Substituting the value of ∠EDC and ∠CDA, we get
x+130∘=180∘
Subtracting 130 on both sides, we get
x=50∘
Hence the value of x is 50∘
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 is an exterior angle of the cyclic quadrilateral ABCD and ∠CBA is the corresponding interior opposite angle, we have
x=∠CBA=50∘
Hence, the value of x is 50∘