In a triangle ABC, `cos 3A+cos 3B+cos3C=1` and `angleA+angleBltangleC`, then find possible measure of `angleC`.

Explanation

To solve the problem, we need to find the possible measure of angle \( C \) in triangle \( ABC \) given that \( \cos 3A + \cos 3B + \cos 3C = 1 \) and \( A + B < C \).

### Step-by-Step Solution:

1. **Understanding the Triangle's Angles**:
In any triangle, the sum of the angles is \( 180^\circ \):
\[ A + B + C = 180^\circ \]

2. **Expressing \( C \)**:
From the equation above, we can express \( C \) in terms of \( A \) and \( B \):
\[ C = 180^\circ - (A + B) \]

3. **Substituting \( C \) into the Cosine Equation**:
We know that \( \cos 3C = \cos(3(180^\circ - (A + B))) \). Using the cosine identity \( \cos(180^\circ - x) = -\cos x \), we have:
\[ \cos 3C = \cos(540^\circ - 3A - 3B) = -\cos(3A + 3B) \]

4. **Rewriting the Given Equation**:
Substitute \( \cos 3C \) into the original equation:
\[ \cos 3A + \cos 3B - \cos(3A + 3B) = 1 \]

5. **Using the Cosine Addition Formula**:
We can use the cosine addition formula:
\[ \cos X + \cos Y = 2 \cos\left(\frac{X + Y}{2}\right) \cos\left(\frac{X - Y}{2}\right) \]
Applying this to \( \cos 3A + \cos 3B \):
\[ \cos 3A + \cos 3B = 2 \cos\left(\frac{3A + 3B}{2}\right) \cos\left(\frac{3A - 3B}{2}\right) \]

6. **Substituting Back**:
Substitute this back into the equation:
\[ 2 \cos\left(\frac{3A + 3B}{2}\right) \cos\left(\frac{3A - 3B}{2}\right) - \cos(3A + 3B) = 1 \]

7. **Setting Up the Equation**:
Let \( x = 3A + 3B \):
\[ 2 \cos\left(\frac{x}{2}\right) \cos\left(\frac{3A - 3B}{2}\right) - \cos x = 1 \]

8. **Using the Identity**:
We can use the identity \( \cos x = 1 - 2 \sin^2\left(\frac{x}{2}\right) \) to rewrite the equation. However, we can also analyze the conditions given.

9. **Finding the Angles**:
Since \( A + B < C \), we have:
\[ A + B < 180^\circ - (A + B) \implies 2(A + B) < 180^\circ \implies A + B < 90^\circ \]

10. **Calculating \( C \)**:
Since \( A + B < 90^\circ \), we can find \( C \):
\[ C = 180^\circ - (A + B) > 180^\circ - 90^\circ = 90^\circ \]

11. **Final Measure of \( C \)**:
Given that the maximum value of \( C \) can be \( 120^\circ \) (as derived from the cosine equation), we conclude:
\[ C = 120^\circ \]

### Conclusion:
The possible measure of angle \( C \) is \( 120^\circ \).