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Question 5: AB is the tangent on the circle at point A. The line BC meets the circle at points C and E. Line AD bisects the ∠ EAC. If ∠ EAC =60 degree and ∠ BAC : ∠ ACB = 2: 5. Find ∠ ABC:

- 40
- 60
- 30
- None of the above

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From the above information we can see that ∠ EAD = ∠ DAC =30 °

And the ratio of ∠ BAC : ∠ ACB = 2 : 5. let's say the ∠ are 2x and 5x

∠ ACB is the exterior ∠ to the triangle AEC, which will be equal to the sum of ∠ EAC and ∠ AEC.

Further, ∠ AED = ∠ BAC (Alternate Segment Theorem!)

**Alternate Segment Theorem: In any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment.**

From here we can plug this information in the equation:

∠ ACB = ∠ EAC + ∠ AEC

5x = 60 ° + 2x

3x = 60 °

x = 20 °

Now, the sum of ∠ in the triangle ACB:

2x + 5x + ∠ ABC = 180 °

So, ∠ ABC = 40 °

The question is **"Find ∠ ABC" **

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