In `Delta PQT, PQ = PT`. The points R and S are on QT such that PR = PS. If `angle PTS = 62^(@) and angle RPS = 34^(@)`, then measure of `angle QPR` is

Options

A. `11^(@)`
B. `13^(@)`
C. `15^(@)`
D. `17^(@)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- In triangle \( \Delta PQT \), we have \( PQ = PT \).
- Points \( R \) and \( S \) are on line \( QT \) such that \( PR = PS \).
- \( \angle PTS = 62^\circ \)
- \( \angle RPS = 34^\circ \)

2. **Use the Isosceles Triangle Property:**
- Since \( PQ = PT \), triangle \( PQT \) is isosceles. Therefore, the angles opposite the equal sides are equal:
\[ \angle PQT = \angle PTQ \]
- Let \( \angle PQT = \angle PTQ = x \).

3. **Calculate \( \angle PQT \):**
- The sum of angles in triangle \( PQT \) is \( 180^\circ \):
\[ \angle PQT + \angle PTQ + \angle PTS = 180^\circ \]
\[ x + x + 62^\circ = 180^\circ \]
\[ 2x + 62^\circ = 180^\circ \]
\[ 2x = 180^\circ - 62^\circ \]
\[ 2x = 118^\circ \]
\[ x = 59^\circ \]
- Thus, \( \angle PQT = \angle PTQ = 59^\circ \).

4. **Consider Triangle \( PRS \):**
- Since \( PR = PS \), triangle \( PRS \) is also isosceles. Therefore:
\[ \angle PRS = \angle PSR \]
- Let \( \angle PRS = \angle PSR = y \).

5. **Calculate \( \angle PRS \):**
- The sum of angles in triangle \( PRS \) is \( 180^\circ \):
\[ \angle PRS + \angle PSR + \angle RPS = 180^\circ \]
\[ y + y + 34^\circ = 180^\circ \]
\[ 2y + 34^\circ = 180^\circ \]
\[ 2y = 180^\circ - 34^\circ \]
\[ 2y = 146^\circ \]
\[ y = 73^\circ \]
- Thus, \( \angle PRS = \angle PSR = 73^\circ \).

6. **Calculate \( \angle QPR \):**
- Now, we can find \( \angle QPR \):
\[ \angle QPR = \angle PRS - \angle PQT \]
\[ \angle QPR = 73^\circ - 59^\circ \]
\[ \angle QPR = 14^\circ \]

### Final Answer:
The measure of \( \angle QPR \) is \( 14^\circ \).