In a cyclic quadrilateral ABCD, `angleA=(x+2)^(@), angleB=(y+3)^(@), angleC=(3y+8)^(@) and angleD=(4x-8)^(@)`. Find the smallest and the largest angle.

Options

A. `48^(@), 143^(@)`
B. `37^(@), 132^(@)`
C. `37^(@), 143^(@)`
D. `20^(@), 132^(@)`

Answer

To solve the problem, we need to use the property of cyclic quadrilaterals which states that the sum of the opposite angles is 180 degrees. Therefore, we can set up the following equations based on the angles given:

1. **Write down the angles:**
- Angle A = \( (x + 2)^\circ \)
- Angle B = \( (y + 3)^\circ \)
- Angle C = \( (3y + 8)^\circ \)
- Angle D = \( (4x - 8)^\circ \)

2. **Set up the equations using the property of cyclic quadrilaterals:**
- From the property, we have:
\[ \text{Angle A} + \text{Angle C} = 180^\circ \]
\[ \text{Angle B} + \text{Angle D} = 180^\circ \]

3. **Substitute the angles into the equations:**
- For Angle A and Angle C:
\[ (x + 2) + (3y + 8) = 180 \]
Simplifying this gives:
\[ x + 3y + 10 = 180 \]
\[ x + 3y = 170 \quad \text{(Equation 1)} \]

- For Angle B and Angle D:
\[ (y + 3) + (4x - 8) = 180 \]
Simplifying this gives:
\[ 4x + y - 5 = 180 \]
\[ 4x + y = 185 \quad \text{(Equation 2)} \]

4. **Now we have a system of equations:**
- Equation 1: \( x + 3y = 170 \)
- Equation 2: \( 4x + y = 185 \)

5. **Solve the system of equations:**
- From Equation 1, express \( x \) in terms of \( y \):
\[ x = 170 - 3y \]
- Substitute this expression for \( x \) into Equation 2:
\[ 4(170 - 3y) + y = 185 \]
\[ 680 - 12y + y = 185 \]
\[ 680 - 11y = 185 \]
\[ -11y = 185 - 680 \]
\[ -11y = -495 \]
\[ y = 45 \]

6. **Substitute \( y \) back to find \( x \):**
- Substitute \( y = 45 \) into Equation 1:
\[ x + 3(45) = 170 \]
\[ x + 135 = 170 \]
\[ x = 35 \]

7. **Now, calculate the angles:**
- Angle A = \( x + 2 = 35 + 2 = 37^\circ \)
- Angle B = \( y + 3 = 45 + 3 = 48^\circ \)
- Angle C = \( 3y + 8 = 3(45) + 8 = 135 + 8 = 143^\circ \)
- Angle D = \( 4x - 8 = 4(35) - 8 = 140 - 8 = 132^\circ \)

8. **Identify the smallest and largest angles:**
- The angles are:
- Angle A = \( 37^\circ \)
- Angle B = \( 48^\circ \)
- Angle C = \( 143^\circ \)
- Angle D = \( 132^\circ \)

- The smallest angle is \( 37^\circ \) (Angle A) and the largest angle is \( 143^\circ \) (Angle C).

### Final Answer:
- **Smallest Angle:** \( 37^\circ \)
- **Largest Angle:** \( 143^\circ \)