In `angle P : angle Q : angle R = 2:2:5.` A line parallel to QR is drawn which touches PQ and RT at A and B respectively. What is the value of `angle PBA - angle PAB` ?

Options

A. 60
B. 30
C. 24
D. 36

Answer

To solve the problem step by step, we will use the properties of angles in triangles and the concept of parallel lines.

### Step 1: Understand the given ratio of angles
We are given that:
\[ \text{angle } P : \text{angle } Q : \text{angle } R = 2 : 2 : 5 \]
Let us denote the angles as:
\[ \text{angle } P = 2x, \quad \text{angle } Q = 2x, \quad \text{angle } R = 5x \]

### Step 2: Use the angle sum property of triangles
The sum of angles in a triangle is 180 degrees. Therefore, we can write:
\[ \text{angle } P + \text{angle } Q + \text{angle } R = 180^\circ \]
Substituting the values we have:
\[ 2x + 2x + 5x = 180^\circ \]
This simplifies to:
\[ 9x = 180^\circ \]

### Step 3: Solve for x
To find the value of \( x \), we divide both sides by 9:
\[ x = \frac{180^\circ}{9} = 20^\circ \]

### Step 4: Calculate the angles
Now we can find the individual angles:
\[ \text{angle } P = 2x = 2 \times 20^\circ = 40^\circ \]
\[ \text{angle } Q = 2x = 2 \times 20^\circ = 40^\circ \]
\[ \text{angle } R = 5x = 5 \times 20^\circ = 100^\circ \]

### Step 5: Analyze the parallel lines
Since line AB is parallel to line QR, we can use the property of alternate interior angles:
- \( \text{angle } PBA = \text{angle } PRQ \)
- \( \text{angle } PAB = \text{angle } PQR \)

### Step 6: Substitute the angles
From our earlier calculations:
\[ \text{angle } PBA = \text{angle } PRQ = 100^\circ \]
\[ \text{angle } PAB = \text{angle } PQR = 40^\circ \]

### Step 7: Calculate \( \text{angle } PBA - \text{angle } PAB \)
Now we can find the difference:
\[ \text{angle } PBA - \text{angle } PAB = 100^\circ - 40^\circ = 60^\circ \]

### Final Answer
Thus, the value of \( \text{angle } PBA - \text{angle } PAB \) is:
\[ \boxed{60^\circ} \]