`PQRST` is a regular pentagon. If PR and QT intersects each other at X, then what is the value (in degrees) of `/_TXR`?

Options

A. 98
B. 90
C. 72
D. 108

Answer

To find the value of angle \( \angle TXR \) in the regular pentagon \( PQRST \) where lines \( PR \) and \( QT \) intersect at point \( X \), we can follow these steps:

### Step 1: Understand the properties of a regular pentagon
A regular pentagon has equal sides and equal angles. The measure of each internal angle in a regular pentagon is given by the formula:
\[ \text{Internal Angle} = \frac{(n-2) \times 180}{n} \]
where \( n \) is the number of sides. For a pentagon (\( n = 5 \)):
\[ \text{Internal Angle} = \frac{(5-2) \times 180}{5} = \frac{3 \times 180}{5} = 108^\circ \]

### Step 2: Identify the relevant triangles
In the pentagon \( PQRST \), we can identify two triangles formed by the intersecting lines \( PR \) and \( QT \):
- Triangle \( PQT \)
- Triangle \( PQR \)

### Step 3: Analyze triangle \( PQT \)
Since \( PQ = PT \) (as sides of the regular pentagon), triangle \( PQT \) is isosceles. Let \( \angle PQT = \angle PTQ = x \). The sum of angles in triangle \( PQT \) gives:
\[ \angle TQP + \angle PQT + \angle PTQ = 180^\circ \]
Substituting the known values:
\[ 108^\circ + x + x = 180^\circ \]
\[ 108^\circ + 2x = 180^\circ \]
\[ 2x = 72^\circ \implies x = 36^\circ \]

### Step 4: Analyze triangle \( PQR \)
Similarly, in triangle \( PQR \), since \( PQ = QR \), it is also isosceles. Let \( \angle QPR = \angle QRP = y \). The sum of angles in triangle \( PQR \) gives:
\[ \angle PQR + \angle QPR + \angle QRP = 180^\circ \]
Substituting the known values:
\[ 108^\circ + y + y = 180^\circ \]
\[ 108^\circ + 2y = 180^\circ \]
\[ 2y = 72^\circ \implies y = 36^\circ \]

### Step 5: Find angle \( \angle TXR \)
Now, we can find \( \angle TXR \) which is equal to \( \angle PXQ \) because of the properties of intersecting lines. We can use the angles found in triangles \( PQT \) and \( PQR \):
\[ \angle TXR = \angle PQT + \angle QPR = 36^\circ + 36^\circ = 72^\circ \]

### Conclusion
Thus, the value of \( \angle TXR \) is:
\[ \angle TXR = 72^\circ \]