Two lines AB and CD intersect each other at a point O such that `angleAOC : angleAOD=5 : 7`. Find all the angles.

Explanation

To solve the problem, we need to find the angles formed by the intersection of two lines AB and CD at point O, given that the ratio of angles AOC to AOD is 5:7.

### Step-by-Step Solution:

1. **Define the Angles**:
Let angle AOC = 5x and angle AOD = 7x, where x is a common multiplier.

**Hint**: Use variables to represent the angles based on the given ratio.

2. **Use the Straight Line Property**:
Since angles AOC and AOD are on a straight line (CD), we know that:
\[ \text{angle AOC} + \text{angle AOD} = 180^\circ \]
Substituting the values we defined:
\[ 5x + 7x = 180^\circ \]

**Hint**: Remember that angles on a straight line sum up to 180 degrees.

3. **Combine Like Terms**:
Combine the terms on the left side:
\[ 12x = 180^\circ \]

**Hint**: Simplify the equation to isolate x.

4. **Solve for x**:
Divide both sides by 12:
\[ x = \frac{180^\circ}{12} = 15^\circ \]

**Hint**: Perform the division carefully to find the value of x.

5. **Calculate the Angles**:
Now, substitute x back into the expressions for the angles:
- Angle AOC:
\[ \text{angle AOC} = 5x = 5 \times 15^\circ = 75^\circ \]
- Angle AOD:
\[ \text{angle AOD} = 7x = 7 \times 15^\circ = 105^\circ \]

**Hint**: Use the value of x to find each angle separately.

6. **Find the Vertically Opposite Angles**:
Since angles AOC and AOD are vertically opposite to angles BOD and COB respectively:
- Angle BOD = Angle AOC = 75°
- Angle COB = Angle AOD = 105°

**Hint**: Recall that vertically opposite angles are equal.

### Final Angles:
- Angle AOC = 75°
- Angle AOD = 105°
- Angle BOD = 75°
- Angle COB = 105°

### Summary:
All angles formed by the intersection of lines AB and CD at point O are:
- Angle AOC = 75°
- Angle AOD = 105°
- Angle BOD = 75°
- Angle COB = 105°