In `DeltaPQR, S and T` are the mid points of sides PQ and PR respectively. If `angle QPR = 45^@ and angle PRQ = 55^@`, then what is the value (in degrees) of `angle QST` ?

Options

A. 80
B. 85
C. 90
D. 100

Answer

To solve the problem step by step, we will analyze the triangle \( \Delta PQR \) and the midpoints \( S \) and \( T \).

### Step 1: Identify the angles in triangle \( PQR \)
Given:
- \( \angle QPR = 45^\circ \)
- \( \angle PRQ = 55^\circ \)

To find \( \angle PQR \), we use the fact that the sum of the angles in a triangle is \( 180^\circ \).

\[ \angle PQR = 180^\circ - \angle QPR - \angle PRQ \]
\[ \angle PQR = 180^\circ - 45^\circ - 55^\circ = 80^\circ \]

### Step 2: Identify the midpoints
Let \( S \) be the midpoint of side \( PQ \) and \( T \) be the midpoint of side \( PR \).

### Step 3: Use the properties of midpoints
Since \( S \) and \( T \) are midpoints, by the Midpoint Theorem, the line segment \( ST \) is parallel to side \( QR \) and \( ST \) is half the length of \( QR \).

### Step 4: Determine the angles involving \( S \) and \( T \)
Since \( ST \parallel QR \), the corresponding angles are equal. Therefore:
\[ \angle QST = \angle PQR = 80^\circ \]

### Step 5: Find \( \angle QST \)
Now, we need to find \( \angle QST \). Since \( S \) and \( T \) are midpoints, we can also analyze triangle \( QST \).

Using the straight line property:
\[ \angle QST + \angle PST = 180^\circ \]
Where \( \angle PST \) is the angle at point \( P \) which is \( \angle QPR = 45^\circ \).

Thus:
\[ \angle QST + 45^\circ = 180^\circ \]
\[ \angle QST = 180^\circ - 45^\circ = 135^\circ \]

However, we need to find \( \angle QST \) in relation to \( \angle PQR \) which we already found to be \( 80^\circ \).

### Final Step: Confirm the angle \( \angle QST \)
Since \( ST \parallel QR \):
\[ \angle QST = \angle PQR = 80^\circ \]

### Conclusion
Thus, the value of \( \angle QST \) is \( 80^\circ \).