In the given figure, if `PQ||ST,anglePQR=110^(@)` and `angleRST = 130^(@)`, then find `angleQRS`.

Answer

To solve the problem, we need to find the measure of angle QRS given that PQ is parallel to ST, angle PQR is 110°, and angle RST is 130°.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- PQ || ST (PQ is parallel to ST)
- ∠PQR = 110°
- ∠RST = 130°

2. **Draw a Parallel Line:**
- Draw a line MN parallel to ST. This gives us three parallel lines: PQ, ST, and MN.

3. **Use the Property of Parallel Lines:**
- Since PQ is parallel to MN, the sum of the interior angles on the same side of the transversal (line QR) must equal 180°.
- Therefore, we can write the equation:
\[ \angle PQR + \angle MRQ = 180° \]

4. **Substitute the Known Angle:**
- Substitute the value of ∠PQR into the equation:
\[ 110° + \angle MRQ = 180° \]

5. **Solve for ∠MRQ:**
- Rearranging the equation gives:
\[ \angle MRQ = 180° - 110° = 70° \]

6. **Use the Other Parallel Line Property:**
- Now, since ST is parallel to MN, we can apply the same property:
\[ \angle RST + \angle NRS = 180° \]

7. **Substitute the Known Angle:**
- Substitute the value of ∠RST into the equation:
\[ 130° + \angle NRS = 180° \]

8. **Solve for ∠NRS:**
- Rearranging gives:
\[ \angle NRS = 180° - 130° = 50° \]

9. **Apply the Linear Pair Property:**
- Now, consider the angles along line MN:
\[ \angle QRM + \angle QRS + \angle SRN = 180° \]
- Substitute the known angles:
\[ 70° + \angle QRS + 50° = 180° \]

10. **Solve for ∠QRS:**
- Combine the known angles:
\[ 120° + \angle QRS = 180° \]
- Rearranging gives:
\[ \angle QRS = 180° - 120° = 60° \]

### Final Answer:
\[ \angle QRS = 60° \]