In a quadrilateral ABCD, ∠A + ∠C = 2(∠B + ∠D). If ∠A = 140° and ∠D = 60°, then ∠B =

Question

Accepted Answer

To solve the problem, we need to determine the value of ∠B in the quadrilateral ABCD, given that ∠A + ∠C = 2(∠B + ∠D) and the values of ∠A and ∠D are 140° and 60°, respectively.

Let's break down the problem step by step:

1. **Given Information:**
   - ∠A = 140°
   - ∠D = 60°
   - ∠A + ∠C = 2(∠B + ∠D)

2. **Substitute the Known Values:**
   - Substitute ∠A = 140° and ∠D = 60° into the equation:
     $$
     140° + ∠C = 2(∠B + 60°)
     $$

3. **Simplify the Equation:**
   - Expand the right side of the equation:
     $$
     140° + ∠C = 2∠B + 120°
     $$

4. **Rearrange the Equation:**
   - Subtract 120° from both sides to isolate terms involving ∠C and ∠B:
     $$
     20° + ∠C = 2∠B
     $$

5. **Use the Sum of Angles in a Quadrilateral:**
   - The sum of angles in any quadrilateral is 360°:
     $$
     ∠A + ∠B + ∠C + ∠D = 360°
     $$
   - Substitute the known values:
     $$
     140° + ∠B + ∠C + 60° = 360°
     $$
   - Simplify:
     $$
     ∠B + ∠C = 160°
     $$

6. **Solve the System of Equations:**
   - We have two equations:
     1. $ 20° + ∠C = 2∠B $
     2. $ ∠B + ∠C = 160° $

- From the second equation, express ∠C in terms of ∠B:
     $$
     ∠C = 160° - ∠B
     $$

- Substitute this expression for ∠C into the first equation:
     $$
     20° + (160° - ∠B) = 2∠B
     $$
   - Simplify:
     $$
     180° - ∠B = 2∠B
     $$
   - Rearrange to solve for ∠B:
     $$
     180° = 3∠B
     $$
     $$
     ∠B = 60°
     $$

7. **Conclusion:**
   - The calculated value of ∠B is 60°, which matches Option 1.

**Explanation for Incorrect Options:**

- **Option 2 ($80^{\circ}$):** If ∠B were 80°, substituting into the equations would not satisfy both the given conditions and the sum of angles in the quadrilateral.
- **Option 3 ($90^{\circ}$):** Similarly, if ∠B were 90°, it would not satisfy the equation 20° + ∠C = 2∠B when checked against the sum of angles.
- **Option 4 ($120^{\circ}$):** If ∠B were 120°, it would exceed the possible sum of angles when combined with the other known angles, violating the quadrilateral angle sum property.

Thus, the correct answer is indeed Option 1, ∠B = 60°.