In `DeltaABC, angleA-angleB=16^(@) and angleC-angleA=34^(@)=34^(@)` , find all angles of the triangle

Answer

To find the angles of triangle ABC given that \( \angle A - \angle B = 16^\circ \) and \( \angle C - \angle A = 34^\circ \), we can follow these steps:

### Step 1: Set up the equations
We know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write:
\[ \angle A + \angle B + \angle C = 180^\circ \]

### Step 2: Express angle C in terms of angle A
From the second condition \( \angle C - \angle A = 34^\circ \), we can express \( \angle C \) as:
\[ \angle C = \angle A + 34^\circ \]

### Step 3: Substitute angle C into the sum of angles equation
Now, substitute \( \angle C \) in the sum of angles equation:
\[ \angle A + \angle B + (\angle A + 34^\circ) = 180^\circ \]
This simplifies to:
\[ 2\angle A + \angle B + 34^\circ = 180^\circ \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 2\angle A + \angle B = 180^\circ - 34^\circ \]
\[ 2\angle A + \angle B = 146^\circ \]

### Step 5: Use the first condition
From the first condition \( \angle A - \angle B = 16^\circ \), we can express \( \angle B \) in terms of \( \angle A \):
\[ \angle B = \angle A - 16^\circ \]

### Step 6: Substitute angle B into the rearranged equation
Substituting \( \angle B \) into the equation \( 2\angle A + \angle B = 146^\circ \):
\[ 2\angle A + (\angle A - 16^\circ) = 146^\circ \]
This simplifies to:
\[ 3\angle A - 16^\circ = 146^\circ \]

### Step 7: Solve for angle A
Adding \( 16^\circ \) to both sides gives:
\[ 3\angle A = 162^\circ \]
Dividing by 3:
\[ \angle A = 54^\circ \]

### Step 8: Find angle B
Now substitute \( \angle A \) back into the equation for \( \angle B \):
\[ \angle B = 54^\circ - 16^\circ = 38^\circ \]

### Step 9: Find angle C
Finally, substitute \( \angle A \) into the equation for \( \angle C \):
\[ \angle C = 54^\circ + 34^\circ = 88^\circ \]

### Conclusion
Thus, the angles of triangle ABC are:
\[ \angle A = 54^\circ, \quad \angle B = 38^\circ, \quad \angle C = 88^\circ \]