In $ riangle$ ABC, if 3∠A = 4∠B = 6∠C, then ∠A is equal to :

Question

In $	riangle$ ABC, if 3∠A = 4∠B = 6∠C, then ∠A is equal to :

Accepted Answer

To solve the problem, we need to determine the measure of angle ∠A in triangle ABC given the relationship 3∠A = 4∠B = 6∠C. We are also given that the correct answer is Option 2: 80°.

### Explanation:

1. **Understanding the Relationship:**
   - We have the relationship 3∠A = 4∠B = 6∠C. Let's denote this common value as k.
   - Therefore, we can express the angles in terms of k:
     - ∠A = k/3
     - ∠B = k/4
     - ∠C = k/6

2. **Sum of Angles in a Triangle:**
   - The sum of angles in any triangle is 180°.
   - Therefore, we have:
     $$
     \frac{k}{3} + \frac{k}{4} + \frac{k}{6} = 180°
     $$

3. **Finding a Common Denominator:**
   - The common denominator for 3, 4, and 6 is 12.
   - Rewriting the equation with a common denominator:
     $$
     \frac{4k}{12} + \frac{3k}{12} + \frac{2k}{12} = 180°
     $$
   - Simplifying, we get:
     $$
     \frac{9k}{12} = 180°
     $$
   - Multiplying both sides by 12 to clear the fraction:
     $$
     9k = 2160
     $$
   - Solving for k:
     $$
     k = \frac{2160}{9} = 240
     $$

4. **Calculating ∠A:**
   - Since ∠A = k/3, we substitute the value of k:
     $$
     ∠A = \frac{240}{3} = 80°
     $$

### Justification for Option 2:

- **Option 2 (80°) is correct** because our calculations show that ∠A is indeed 80° when we solve the given relationship and apply the sum of angles in a triangle.

### Explanation for Incorrect Options:

- **Option 1 (60°):** This would imply a different value of k that does not satisfy the equation derived from the sum of angles in the triangle.
- **Option 3 (30°):** This is too small and would not satisfy the equation derived from the sum of angles in the triangle.
- **Option 4 (40°):** This is also incorrect as it does not satisfy the derived equation for the sum of angles.

In conclusion, based on the given relationships and the sum of angles in a triangle, Option 2 (80°) is the correct answer for ∠A.