In a cyclic -quadrilateral PQRS angle PQR ` = 135^(@) ` , Sides SP and RQ produced meet at point A whereas sides PQ and SR produced meet at point B. If ` angle A : angle B = 2: 1`. find angles A and B

Answer

To solve the problem, we need to find angles A and B in the cyclic quadrilateral PQRS, given that angle PQR = 135° and the ratio of angles A to B is 2:1.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Angle PQR = 135°
- The ratio of angles A to B = 2:1

2. **Use the Property of Cyclic Quadrilaterals**:
- In a cyclic quadrilateral, the sum of opposite angles is 180°. Therefore, we can find angle S:
\[ \text{Angle S} + \text{Angle Q} = 180° \]
Since angle Q is angle PQR, we have:
\[ \text{Angle S} + 135° = 180° \]
\[ \text{Angle S} = 180° - 135° = 45° \]

3. **Determine Angle PQA**:
- Since SP and RQ are produced to meet at point A, angle PQA is supplementary to angle PQR:
\[ \text{Angle PQA} + \text{Angle PQR} = 180° \]
\[ \text{Angle PQA} + 135° = 180° \]
\[ \text{Angle PQA} = 180° - 135° = 45° \]

4. **Identify Angles at Point B**:
- Angle A and angle B are formed by the intersection of the extended lines at points A and B. Since angle PQA = 45°, angle A can be expressed in terms of angle B:
\[ \text{Angle SRQ} = \text{Angle S} + \text{Angle B} = 45° + \text{Angle B} \]

5. **Set Up the Equation Using the Ratio**:
- Let angle B = x. Then angle A = 2x. The sum of angles around point B gives:
\[ 45° + 45° + x + 2x = 180° \]
Simplifying this, we have:
\[ 90° + 3x = 180° \]
\[ 3x = 180° - 90° = 90° \]
\[ x = 30° \]

6. **Calculate Angles A and B**:
- Angle B = x = 30°
- Angle A = 2x = 2(30°) = 60°

### Final Answer:
- Angle A = 60°
- Angle B = 30°