In the given figure., AB and CD are two lines intersecting at O. Calculate the following-angle AOD,ANGLE AOC,ANGLE BOD AND ANGLE BOC IF ANGLE AOD :AOC=5:4

To solve this problem, let's follow these steps:

### Step 1: Define the Angles
Let's assume the measure of angle \( \angle AOD \) is \( 5x \), and the measure of angle \( \angle AOC \) is \( 4x \), as per the given ratio \( 5:4 \).

### Step 2: Use the Property of Straight Line
Since \( AB \) and \( CD \) intersect at point \( O \), the angles \( \angle AOD \) and \( \angle AOC \) are adjacent angles along the straight line \( AC \). Therefore, their sum will be \( 180^\circ \):
\[
\angle AOD + \angle AOC = 180^\circ
\]
Substituting the expressions:
\[
5x + 4x = 180^\circ
\]
\[
9x = 180^\circ
\]
\[
x = \frac{180^\circ}{9} = 20^\circ
\]

### Step 3: Calculate Each Angle
- \( \angle AOD = 5x = 5 \times 20^\circ = 100^\circ \)
- \( \angle AOC = 4x = 4 \times 20^\circ = 80^\circ \)

### Step 4: Use Vertical Angles
Since \( AB \) and \( CD \) intersect at \( O \), the angles \( \angle AOD \) and \( \angle BOC \) are vertical angles and thus equal. Similarly, \( \angle AOC \) and \( \angle BOD \) are vertical angles and thus equal.
\[
\angle BOC = \angle AOD = 100^\circ
\]
\[
\angle BOD = \angle AOC = 80^\circ
\]

### Final Answers:
- \( \angle AOD = 100^\circ \)
- \( \angle AOC = 80^\circ \)
- \( \angle BOD = 80^\circ \)
- \( \angle BOC = 100^\circ \)

Would you like more details on how these properties work, or do you have any other questions?

### Relative Questions:
1. What are vertical angles and how do they relate in intersecting lines?
2. How do we determine the sum of angles on a straight line?
3. How can you solve problems involving angle ratios?
4. What is the relationship between adjacent angles?
5. How do intersecting lines affect the measurement of angles?

### Tip:
When dealing with angle problems involving intersecting lines, remember that vertical angles are always equal, and the sum of adjacent angles on a straight line is always \( 180^\circ \).

Learn how to calculate angles AOD, AOC, BOD, and BOC in intersecting lines problem. Solve using the angle ratio 5:4 with step-by-step explanations. Suitable for Grades 7-9 students and anyone interested in geometry.

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