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

Q: In $\triangle ΑBC$, if 3∠A = 4∠B = 6∠C, then ∠A is equal to :

In $\triangle ΑBC$, if 3∠A = 4∠B = 6∠C, then ∠A is equal to :

Question

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

Accepted Answer

To solve the problem, we need to find the measure of angle ∠A in triangle ABC, given that 3∠A = 4∠B = 6∠C. We are also given that the correct answer is Option 1: 80°.

### Explanation:

1. **Understanding the Relationship:**
   - We have the relationships: 3∠A = 4∠B = 6∠C.
   - Let this common value be denoted by x. Therefore, we have:
     - 3∠A = x
     - 4∠B = x
     - 6∠C = x

2. **Expressing Angles in Terms of x:**
   - From 3∠A = x, we get ∠A = x/3.
   - From 4∠B = x, we get ∠B = x/4.
   - From 6∠C = x, we get ∠C = x/6.

3. **Using the Angle Sum Property of a Triangle:**
   - The sum of angles in a triangle is always 180°.
   - Therefore, ∠A + ∠B + ∠C = 180°.
   - Substituting the expressions in terms of x, we get:
     $$
     \frac{x}{3} + \frac{x}{4} + \frac{x}{6} = 180°
     $$

4. **Solving the Equation:**
   - To solve the equation, find a common denominator for the fractions, which is 12:
     $$
     \frac{4x}{12} + \frac{3x}{12} + \frac{2x}{12} = 180°
     $$
   - Combine the fractions:
     $$
     \frac{4x + 3x + 2x}{12} = 180°
     $$
     $$
     \frac{9x}{12} = 180°
     $$
   - Simplify the fraction:
     $$
     \frac{3x}{4} = 180°
     $$
   - Multiply both sides by 4 to clear the fraction:
     $$
     3x = 720°
     $$
   - Divide by 3 to solve for x:
     $$
     x = 240°
     $$

5. **Finding ∠A:**
   - Recall that ∠A = x/3.
   - Substitute x = 240°:
     $$
     ∠A = \frac{240°}{3} = 80°
     $$

### Conclusion:

- **Option 1 (80°) is correct** because, based on the given relationships and the angle sum property of a triangle, ∠A is calculated to be 80°.

### Explanation for Incorrect Options:

- **Option 2 (60°):** This would imply a different value of x, which does not satisfy the equation derived from the angle sum property.
- **Option 3 (40°):** Similarly, this would require a different x value, leading to an incorrect total angle sum.
- **Option 4 (30°):** This option also results in an incorrect angle sum, as it does not satisfy the derived equation.

Thus, the correct answer is indeed Option 1: 80°.