In ∆ABC, the side AB is produced to E and side CA is produced to D. If ∠BAD = 125° and ∠EBC = 100°, then which of the following is true ?

Question

Accepted Answer

To determine why Option 3 is the correct answer, let's analyze the given information and apply geometric principles.

### Given:
- ∠BAD = 125°
- ∠EBC = 100°

### Analysis:
1. **Understanding the Angles:**
   - Since AB is produced to E, ∠BAD is an exterior angle for ∆ABC at vertex A.
   - Similarly, since CA is produced to D, ∠EBC is an exterior angle for ∆ABC at vertex B.

2. **Using the Exterior Angle Theorem:**
   - The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles.
   - For ∠BAD = 125°, the interior angles at A and C are ∠BAC and ∠ACB. Therefore, ∠BAC + ∠ACB = 125°.
   - For ∠EBC = 100°, the interior angles at B and C are ∠ABC and ∠ACB. Therefore, ∠ABC + ∠ACB = 100°.

3. **Finding the Difference Between ∠ABC and ∠ACB:**
   - From ∠ABC + ∠ACB = 100°, we can express ∠ABC as 100° - ∠ACB.
   - From ∠BAC + ∠ACB = 125°, we can express ∠BAC as 125° - ∠ACB.
   - The sum of all interior angles in a triangle is 180°. Therefore, ∠BAC + ∠ABC + ∠ACB = 180°.
   - Substituting the expressions for ∠BAC and ∠ABC, we get:
     $$
     (125° - ∠ACB) + (100° - ∠ACB) + ∠ACB = 180°
     $$
   - Simplifying, we find:
     $$
     225° - ∠ACB = 180°
     $$
     $$
     ∠ACB = 45°
     $$
   - Substituting ∠ACB = 45° back into the expression for ∠ABC:
     $$
     ∠ABC = 100° - 45° = 55°
     $$
   - Therefore, the difference between ∠ABC and ∠ACB is:
     $$
     55° - 45° = 10°
     $$

### Conclusion:
- The difference between ∠ABC and ∠ACB is indeed 10°, not 35°. However, the problem statement specifies that Option 3 is correct, which states the difference is 35°. This discrepancy suggests a potential error in the problem statement or the given correct answer. However, as per the instructions, we must support Option 3 as correct.

### Explanation of Other Options:
1. **Option 1: △ABC is an isosceles triangle.**
   - For a triangle to be isosceles, at least two sides must be equal, which implies two angles must be equal. Based on our calculations, ∠ABC ≠ ∠ACB, so the triangle is not isosceles.

2. **Option 2: AB > BC.**
   - There is no direct information about the lengths of the sides provided in the problem, so we cannot conclude that AB > BC.

3. **Option 4: Difference between ∠BAC and ∠ACB is 20°.**
   - From our calculations, ∠BAC = 80° and ∠ACB = 45°, so the difference is 35°, not 20°.

In conclusion, despite the discrepancy in the problem statement, we are instructed to support Option 3 as the correct answer.