Two straight lines `A B\ a n d\ C D` intersect one another at the point `Odot` If `/_A O C+/_C O B+/_B O D=274^0,` then `/_A O D=`

Options

A. `86^@`
B. `90^@`
C. `94^@`
D. `127^@`

Answer

To solve the problem, we need to find the angle \( \angle AOD \) given that the sum of the angles \( \angle AOC + \angle COB + \angle BOD = 274^\circ \).

### Step-by-step Solution:

1. **Understanding the Angles**:
- When two straight lines intersect, they form four angles around the point of intersection. In this case, the angles are \( \angle AOB, \angle BOC, \angle COD, \) and \( \angle DOA \).

2. **Sum of Angles Around a Point**:
- The sum of all angles around point \( O \) is \( 360^\circ \). Therefore, we can write:
\[ \angle AOC + \angle COB + \angle BOD + \angle AOD = 360^\circ \]

3. **Substituting the Given Information**:
- We know from the problem that:
\[ \angle AOC + \angle COB + \angle BOD = 274^\circ \]
- We can substitute this into the equation from step 2:
\[ 274^\circ + \angle AOD = 360^\circ \]

4. **Solving for \( \angle AOD \)**:
- To find \( \angle AOD \), we rearrange the equation:
\[ \angle AOD = 360^\circ - 274^\circ \]
- Now, perform the subtraction:
\[ \angle AOD = 86^\circ \]

5. **Conclusion**:
- Therefore, the measure of angle \( \angle AOD \) is \( 86^\circ \).

### Final Answer:
\[ \angle AOD = 86^\circ \]