If O be the circumcentre of a triangle `PQR `and `/_QOR = 110^@, /_OPR = 25^@`, then the measure of `/_PRQ` is

Options

A. `65^@`
B. `50^@`
C. `55^@`
D. ` 60^@`

Answer

To find the measure of angle \( \angle PRQ \) in triangle \( PQR \) given that \( O \) is the circumcenter, \( \angle QOR = 110^\circ \), and \( \angle OPR = 25^\circ \), we can follow these steps:

### Step 1: Understand the relationship between angles at the circumcenter
The circumcenter \( O \) of triangle \( PQR \) is the point where the perpendicular bisectors of the sides intersect. The angles formed at the circumcenter relate to the angles of the triangle.

### Step 2: Use the property of angles at the circumcenter
The angle \( \angle QOR \) is related to the angles \( \angle PQR \) and \( \angle PRQ \) as follows:
\[ \angle QOR = 2 \cdot \angle PRQ \]
This is because the angle at the circumcenter is twice the angle at the opposite vertex.

### Step 3: Calculate \( \angle PRQ \)
Given \( \angle QOR = 110^\circ \):
\[ 110^\circ = 2 \cdot \angle PRQ \]
To find \( \angle PRQ \), we divide both sides by 2:
\[ \angle PRQ = \frac{110^\circ}{2} = 55^\circ \]

### Step 4: Find \( \angle PQR \)
Next, we need to find \( \angle PQR \). We know that the angles in triangle \( PQR \) must sum to \( 180^\circ \):
\[ \angle PQR + \angle PRQ + \angle OPR = 180^\circ \]
Substituting the known values:
\[ \angle PQR + 55^\circ + 25^\circ = 180^\circ \]
This simplifies to:
\[ \angle PQR + 80^\circ = 180^\circ \]
Subtracting \( 80^\circ \) from both sides gives:
\[ \angle PQR = 180^\circ - 80^\circ = 100^\circ \]

### Step 5: Conclusion
Thus, we have found that:
\[ \angle PRQ = 55^\circ \]

### Final Answer
The measure of \( \angle PRQ \) is \( 55^\circ \).

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