`/_A, /_B, /_C `are three angles of a triangle. If `/_A - /_B = 15^@, /_B - /_C = 30^@`, then `/_A, /_B and /_C` are

Options

A. `80^@, 60^@, 40^@`
B. `70^@, 50^@, 60^@`
C. `80^@, 65^@, 35^@`
D. `80^@, 55^@, 45^@`

Answer

To solve for the angles \( \angle A \), \( \angle B \), and \( \angle C \) of the triangle given the conditions \( \angle A - \angle B = 15^\circ \) and \( \angle B - \angle C = 30^\circ \), we can follow these steps:

### Step 1: Set up the equations
We start with the two given equations:
1. \( \angle A - \angle B = 15^\circ \) (Equation 1)
2. \( \angle B - \angle C = 30^\circ \) (Equation 2)

### Step 2: Express angles in terms of \( \angle B \)
From Equation 1, we can express \( \angle A \) in terms of \( \angle B \):
\[ \angle A = \angle B + 15^\circ \quad \text{(Equation 3)} \]

From Equation 2, we can express \( \angle C \) in terms of \( \angle B \):
\[ \angle C = \angle B - 30^\circ \quad \text{(Equation 4)} \]

### Step 3: Use the triangle angle sum property
The sum of the angles in a triangle is \( 180^\circ \):
\[ \angle A + \angle B + \angle C = 180^\circ \quad \text{(Equation 5)} \]

### Step 4: Substitute Equations 3 and 4 into Equation 5
Substituting \( \angle A \) and \( \angle C \) from Equations 3 and 4 into Equation 5:
\[ (\angle B + 15^\circ) + \angle B + (\angle B - 30^\circ) = 180^\circ \]

### Step 5: Simplify the equation
Combine like terms:
\[ 3\angle B - 15^\circ = 180^\circ \]

### Step 6: Solve for \( \angle B \)
Add \( 15^\circ \) to both sides:
\[ 3\angle B = 195^\circ \]
Now divide by 3:
\[ \angle B = 65^\circ \]

### Step 7: Find \( \angle A \) and \( \angle C \)
Now that we have \( \angle B \), we can find \( \angle A \) and \( \angle C \) using Equations 3 and 4:
\[ \angle A = \angle B + 15^\circ = 65^\circ + 15^\circ = 80^\circ \]
\[ \angle C = \angle B - 30^\circ = 65^\circ - 30^\circ = 35^\circ \]

### Final Result
Thus, the angles of the triangle are:
- \( \angle A = 80^\circ \)
- \( \angle B = 65^\circ \)
- \( \angle C = 35^\circ \)

### Summary
The angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( 80^\circ \), \( 65^\circ \), and \( 35^\circ \) respectively.

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