In the below figure, O is the centre of the circle. What is angle ABC?

Question

https://gmatclub.com/forum/download/file.php?id=46342 In the below figure, O is the centre of the circle. What is angle ABC? (1) Angle OAC = 40 Degrees (2) Angle AOC = 100 Degrees Circle.jpg

4d · Answer

The sum of the opposite angles of a cyclic quadrilateral is supplementary.

(1) Angle OAC = 40 Degrees
Angle OAC = Angle OCA  
Hence AOC = 100
ABC = 80
Sufficient

(2)AOC = 100
ABC = 80
Sufficient 
IMO D

Archit3110 · Answer

#1
OA = OC= 40
Angle AOC= 100
central angle = 0.5 of angle subtended on 
insribed angle is equal to half of the central angle it intercepts
sufficient
#2
again AOC = 100
so 50= ABC
IMO D

VictorFeng · Answer

The inscribed angle ABC should be half of the intercepted arc.
If angle AOC = 100, then intercepted arc is 360 - 100 = 260
Angle ABC is 1/2 * 260 = 130.

Answer D

Manikanta047 · Answer

We are given that O is the center of the circle, and A , B , C , and O form a quadrilateral inside the circle. We are also given \angle ACO = 40^\circ , and we are trying to find \angle ABC .

Reanalysis of the problem:

1. Angle ACO :
Given \angle ACO = 40^\circ .
 2. Central angle relationship:
 O is the center of the circle, and OA and OC are radii of the circle. So, triangle OAC is isosceles. But in this problem, the focus is on the angles related to the quadrilateral ABCO .
 3. Cyclic Quadrilateral Property:
The quadrilateral ABCO is inscribed in the circle. One key property of cyclic quadrilaterals is that opposite angles are supplementary (their sum is 180^\circ ).
 4. Finding \angle ABC :
You are given that \angle ACO = 40^\circ . Since \angle ABC and \angle ACO are opposite angles in the cyclic quadrilateral, they must add up to 180^\circ :

\angle ABC + \angle ACO = 180^\circ

Therefore:

\angle ABC = 180^\circ - 40^\circ = 140^\circ

Conclusion:

So, the correct measure of \angle ABC is 140 degrees, based on the cyclic quadrilateral property, not 130 degrees.

In the below figure, O is the centre of the circle. What is angle ABC?

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GMAT Data Sufficiency (DS) Questions

In the below figure, O is the centre of the circle. What is angle ABC?

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