In a quadrilateral PQRS, angle Q is twice the angle P. Angle R is thrice the angle P. The value of angle R is `150^@` What is the difference between angle Q and angle S?

Options

A. `30^@`
B. `60^@`
C. `40^@`
D. `35^@`

Answer

Let's solve the problem step by step.

### Step 1: Define the angles in terms of angle P
Let the measure of angle P be \( x \). According to the problem:
- Angle Q is twice angle P, so \( Q = 2x \).
- Angle R is thrice angle P, so \( R = 3x \).

### Step 2: Use the given value of angle R
We know from the problem that angle R is \( 150^\circ \). Therefore, we can set up the equation:
\[ 3x = 150 \]

### Step 3: Solve for angle P
To find the value of \( x \), we divide both sides of the equation by 3:
\[ x = \frac{150}{3} = 50 \]

### Step 4: Calculate angles Q and S
Now that we have \( x \), we can find angle Q:
\[ Q = 2x = 2 \times 50 = 100^\circ \]

Next, we need to find angle S. The sum of the angles in a quadrilateral is \( 360^\circ \). Therefore, we can write:
\[ P + Q + R + S = 360 \]
Substituting the values we have:
\[ 50 + 100 + 150 + S = 360 \]

### Step 5: Solve for angle S
Now, simplify the equation:
\[ 300 + S = 360 \]
Subtracting 300 from both sides gives:
\[ S = 360 - 300 = 60^\circ \]

### Step 6: Find the difference between angles Q and S
Now, we can find the difference between angle Q and angle S:
\[ Q - S = 100^\circ - 60^\circ = 40^\circ \]

### Final Answer
The difference between angle Q and angle S is \( 40^\circ \).

---