In the Below Figure, O is The Centre of The Circle GMAT Data Sufficiency

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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“In the below figure, O is the centre of the circle. What is angle ABC?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide 2022". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

Remember one rule

A cyclic quadrilateral's opposite angles added together form a supplementary angle.
Looking at statement 1
Angle OAC = 40 Degrees
Angle OAC = Angle OCA [radius]
Therefore AOC = 100
ABC = 80
Hence This is Sufficient.
Looking at statement 2
Angle AOC = 100 Degrees
From statement 1 we already know
AOC = 100
ABC = 80
Hence This is Sufficient.

EACH statement ALONE is sufficient.

Correct Answer: D

Approach Solution 2:
There is another approach to solve the question
Looking at statement 1
Angle OAC = 40 Degrees
OA = OC= 40
Angle AOC= 100
Half of the central angle it intercepts is equal to the angle subtended on the inscribed angle, which is the central angle.
Hence This is Sufficient.
Looking at statement 2
Angle AOC = 100 Degrees
From statement 1 we already know
Half of the central angle it intercepts is equal to the angle subtended on the inscribed angle, which is the central angle.
again AOC = 100
so ABC = 50
Hence This is Sufficient.

EACH statement ALONE is sufficient.

Correct Answer: D

Approach Solution 3:

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

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

(2)AOC = 100
ABC = 80
Sufficient

EACH statement ALONE is sufficient.

Correct Answer: D

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