Angle 'A' of the quadrilateral ABCD is `26^(@)` less than angle B. Angle B is twice angle C and angle C is `10^(@)` more than the angle D. What would be the measure of angle A?

Options

A. `104^(@)`
B. `126^(@)`
C. `56^(@)`
D. `106^(@)`

Answer

To find the measure of angle A in quadrilateral ABCD, we can set up equations based on the relationships given in the problem.

1. **Define Variables:**
- Let angle D = x (in degrees).
- Then, angle C = x + 10 (since angle C is 10 degrees more than angle D).
- Angle B = 2 * angle C = 2(x + 10) = 2x + 20 (since angle B is twice angle C).
- Angle A = angle B - 26 = (2x + 20) - 26 = 2x - 6 (since angle A is 26 degrees less than angle B).

2. **Sum of Angles in Quadrilateral:**
- The sum of the angles in a quadrilateral is 360 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360 \]
- Substituting the expressions we have:
\[ (2x - 6) + (2x + 20) + (x + 10) + x = 360 \]

3. **Combine Like Terms:**
- Combine the terms:
\[ 2x - 6 + 2x + 20 + x + 10 + x = 360 \]
- This simplifies to:
\[ 6x + 24 = 360 \]

4. **Solve for x:**
- Subtract 24 from both sides:
\[ 6x = 360 - 24 \]
\[ 6x = 336 \]
- Divide by 6:
\[ x = \frac{336}{6} = 56 \]

5. **Find the Angles:**
- Now that we have x, we can find the measures of the angles:
- Angle D = x = 56 degrees
- Angle C = x + 10 = 56 + 10 = 66 degrees
- Angle B = 2(x + 10) = 2(66) = 132 degrees
- Angle A = 2x - 6 = 2(56) - 6 = 112 - 6 = 106 degrees

6. **Conclusion:**
- Therefore, the measure of angle A is **106 degrees**.