The angles of a triangle are \(x\), \(y\), and \(40^\circ\). Their difference \(|x - y|\) is \(30^\circ\). Find \(x\) and \(y\).
\(x = 85^\circ\), \(y = 55^\circ\) (or vice versa).
Step 1: In any triangle, the sum of the three angles is always \(180^\circ\).
So,
\(x + y + 40 = 180\)
\(x + y = 180 - 40\)
\(x + y = 140\)
Step 2: We are also told that the difference between \(x\) and \(y\) is \(30^\circ\).
This means:
\(|x - y| = 30\)
Step 3: Now we have two equations:
1) \(x + y = 140\)
2) \(|x - y| = 30\)
Step 4: Let us assume \(x > y\). Then \(x - y = 30\).
Step 5: Add the two equations:
\((x + y) + (x - y) = 140 + 30\)
\(2x = 170\)
\(x = 85\)
Step 6: Put \(x = 85\) in equation (1):
\(85 + y = 140\)
\(y = 55\)
Step 7: If we had assumed \(y > x\), then we would get \(y = 85\) and \(x = 55\).
Final Answer: The two angles are \(85^\circ\) and \(55^\circ\).