In `triangleABC and trianglePQR, AB = PQ, angleA = angleP` and `angleB = angleQ`. If `angleA + angleC = 130^(@)`, then `angleQ` = ..........

Options

A. `65^(@)`
B. `130^(@)`
C. `50^(@)`
D. `100^(@)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that:
- In triangle ABC, \( AB = PQ \)
- \( \angle A = \angle P \)
- \( \angle B = \angle Q \)
- \( \angle A + \angle C = 130^\circ \)

### Step 2: Use the triangle angle sum property
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, for triangle ABC:
\[ \angle A + \angle B + \angle C = 180^\circ \]

### Step 3: Substitute the known values
From the information given, we can substitute \( \angle C \) using \( \angle A + \angle C = 130^\circ \):
\[ \angle C = 130^\circ - \angle A \]

Now, substituting this into the triangle angle sum equation:
\[ \angle A + \angle B + (130^\circ - \angle A) = 180^\circ \]

### Step 4: Simplify the equation
By simplifying the equation:
\[ \angle B + 130^\circ = 180^\circ \]
\[ \angle B = 180^\circ - 130^\circ \]
\[ \angle B = 50^\circ \]

### Step 5: Relate angle B to angle Q
Since it is given that \( \angle B = \angle Q \), we conclude:
\[ \angle Q = 50^\circ \]

### Final Answer
Thus, the value of \( \angle Q \) is \( 50^\circ \).

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