Ray AX is the bisector of `angleBAC` and ray AY is the bisector of `angleXAC`. If `angleBAY=60^(@)`, find `angleBAC`.

Answer

To solve the problem step by step, we will follow the logical deductions based on the information provided in the question.

### Step-by-Step Solution:

1. **Understanding the Angles**:
- Let the measure of angle \( BAC \) be \( x \).
- Since ray \( AX \) is the bisector of angle \( BAC \), it divides \( BAC \) into two equal parts. Therefore, \( \angle BAX = \frac{x}{2} \) and \( \angle XAC = \frac{x}{2} \).

2. **Identifying the Second Bisector**:
- Ray \( AY \) is the bisector of angle \( XAC \). Since \( \angle XAC = \frac{x}{2} \), ray \( AY \) divides this angle into two equal parts:
- \( \angle XAY = \frac{1}{2} \times \frac{x}{2} = \frac{x}{4} \)
- Therefore, \( \angle YAC = \frac{x}{4} \) as well.

3. **Setting Up the Equation**:
- We know from the problem that \( \angle BAY = 60^\circ \).
- We can express \( \angle BAY \) in terms of \( x \):
\[ \angle BAY = \angle BAX + \angle XAY = \frac{x}{2} + \frac{x}{4} \]

4. **Finding a Common Denominator**:
- To add \( \frac{x}{2} \) and \( \frac{x}{4} \), we need a common denominator. The least common multiple of 2 and 4 is 4:
\[ \frac{x}{2} = \frac{2x}{4} \]
- Now we can add:
\[ \angle BAY = \frac{2x}{4} + \frac{x}{4} = \frac{3x}{4} \]

5. **Setting Up the Equation**:
- Since we know \( \angle BAY = 60^\circ \), we can set up the equation:
\[ \frac{3x}{4} = 60 \]

6. **Solving for \( x \)**:
- To solve for \( x \), multiply both sides by 4:
\[ 3x = 240 \]
- Now, divide both sides by 3:
\[ x = \frac{240}{3} = 80 \]

7. **Conclusion**:
- Therefore, the measure of angle \( BAC \) is:
\[ \angle BAC = x = 80^\circ \]

### Final Answer:
\[ \angle BAC = 80^\circ \]