Let `veca, vecb and vecc` are three non - collinear vectors in a plane such that `|veca|=2, |vecb|=5 and |vecc|=sqrt(29)`. If the angle between `veca and vecc` is `theta_(1)` and the angle between `vecb and vecc` is `theta_(1)` the angle between `vecb and vecc` is `theta_(2)`, where `theta_(1), theta_(2) in [(pi)/(2),pi]`, then the value of `theta_(1)+theta_(2)` is equal to

Options

A. `(7pi)/(6)`
B. `(4pi)/(6)`
C. `(3pi)/(2)`
D. `(7pi)/(4)`

Answer

To solve the problem, we need to analyze the given vectors and their relationships based on the information provided.

### Step-by-step solution:

1. **Understanding the Vectors**:
We have three non-collinear vectors \( \vec{a}, \vec{b}, \) and \( \vec{c} \) with magnitudes:
- \( |\vec{a}| = 2 \)
- \( |\vec{b}| = 5 \)
- \( |\vec{c}| = \sqrt{29} \)

2. **Using the Law of Cosines**:
The law of cosines states that for any triangle with sides \( a, b, c \) opposite to angles \( A, B, C \):
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
Here, we can apply this to find the angles \( \theta_1 \) and \( \theta_2 \).

3. **Finding \( \theta_1 \)**:
For the angle \( \theta_1 \) between \( \vec{a} \) and \( \vec{c} \):
\[ |\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 |\vec{a}| |\vec{b}| \cos(\theta_1) \]
Substituting the values:
\[ 29 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cos(\theta_1) \]
\[ 29 = 4 + 25 - 20 \cos(\theta_1) \]
\[ 29 = 29 - 20 \cos(\theta_1) \]
This simplifies to:
\[ 0 = -20 \cos(\theta_1) \implies \cos(\theta_1) = 0 \]
Therefore, \( \theta_1 = \frac{\pi}{2} \).

4. **Finding \( \theta_2 \)**:
Similarly, for the angle \( \theta_2 \) between \( \vec{b} \) and \( \vec{c} \):
\[ |\vec{c}|^2 = |\vec{b}|^2 + |\vec{a}|^2 - 2 |\vec{b}| |\vec{a}| \cos(\theta_2) \]
Substituting the values:
\[ 29 = 5^2 + 2^2 - 2 \cdot 5 \cdot 2 \cos(\theta_2) \]
\[ 29 = 25 + 4 - 20 \cos(\theta_2) \]
\[ 29 = 29 - 20 \cos(\theta_2) \]
This simplifies to:
\[ 0 = -20 \cos(\theta_2) \implies \cos(\theta_2) = 0 \]
Therefore, \( \theta_2 = \frac{\pi}{2} \).

5. **Finding \( \theta_1 + \theta_2 \)**:
Now we can find the sum of the angles:
\[ \theta_1 + \theta_2 = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

### Final Answer:
The value of \( \theta_1 + \theta_2 \) is \( \pi \).