In a `/_\ABC`, if `A-B = 33^(@)` and `B-C = 18^(@)`, then `/_B = ?`

Options

A. `35^(@)`
B. `55^(@)`
C. `45^(@)`
D. `57^(@)`

Answer

To solve the problem, we need to find the measure of angle B in triangle ABC given the relationships between the angles. Here’s a step-by-step solution:

### Step 1: Define the angles in terms of angle B
We are given two relationships:
1. \( A - B = 33^\circ \)
2. \( B - C = 18^\circ \)

From the first equation, we can express angle A in terms of angle B:
\[ A = B + 33^\circ \]

From the second equation, we can express angle C in terms of angle B:
\[ C = B - 18^\circ \]

### Step 2: Use the triangle angle sum property
In any triangle, the sum of the interior angles is always \( 180^\circ \). Therefore, we can write:
\[ A + B + C = 180^\circ \]

Substituting the expressions for A and C from Step 1:
\[ (B + 33^\circ) + B + (B - 18^\circ) = 180^\circ \]

### Step 3: Simplify the equation
Combining like terms:
\[ B + 33^\circ + B + B - 18^\circ = 180^\circ \]
This simplifies to:
\[ 3B + 15^\circ = 180^\circ \]

### Step 4: Solve for angle B
Now, isolate \( 3B \):
\[ 3B = 180^\circ - 15^\circ \]
\[ 3B = 165^\circ \]

Now, divide both sides by 3:
\[ B = \frac{165^\circ}{3} \]
\[ B = 55^\circ \]

### Conclusion
The measure of angle B is \( 55^\circ \).

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