In a triangle `PQR, PQ = PR` and `/_Q` is twice that of `/_P`. Then `/_Q` Is equal to

Options

A. `72^@`
B. `36^@`
C. `144^@`
D. `108^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Triangle Properties
We have a triangle \( PQR \) where \( PQ = PR \). This means that triangle \( PQR \) is an isosceles triangle, and the angles opposite to equal sides are equal.

### Step 2: Define the Angles
Let \( \angle P = \theta \). Since \( \angle Q \) is twice that of \( \angle P \), we can write:
\[ \angle Q = 2\theta \]
Since \( PQ = PR \), we also have:
\[ \angle R = \theta \]

### Step 3: Use the Triangle Sum Property
The sum of the angles in any triangle is \( 180^\circ \). Therefore, we can write the equation:
\[ \angle P + \angle Q + \angle R = 180^\circ \]
Substituting the angles we defined:
\[ \theta + 2\theta + \theta = 180^\circ \]

### Step 4: Simplify the Equation
Combine the terms:
\[ 4\theta = 180^\circ \]

### Step 5: Solve for \( \theta \)
To find \( \theta \), divide both sides by 4:
\[ \theta = \frac{180^\circ}{4} = 45^\circ \]

### Step 6: Find \( \angle Q \)
Now that we have \( \theta \), we can find \( \angle Q \):
\[ \angle Q = 2\theta = 2 \times 45^\circ = 90^\circ \]

### Final Answer
Thus, \( \angle Q \) is equal to \( 90^\circ \).

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