In the given figure,AB and CD are two lines intersecting at O.calculate the following- angle AOD,angle AOX, angle BOD,and Angle BOC if angle AOD: AOC=5:4

Given that AB and CD are two lines intersecting at point O, and the ratio of angles \( \angle AOD : \angle AOC \) is 5:4, we need to calculate the angles \( \angle AOD \), \( \angle AOX \), \( \angle BOD \), and \( \angle BOC \).

### Step 1: Understanding the Ratio
Let the angles \( \angle AOD \) and \( \angle AOC \) be represented as 5x and 4x respectively, where x is a common factor.

### Step 2: Sum of Angles around a Point
Since angles \( \angle AOD \), \( \angle AOC \), \( \angle BOC \), and \( \angle BOD \) form a complete circle around point O, their sum is \( 360^\circ \).

\[ \angle AOD + \angle AOC + \angle BOC + \angle BOD = 360^\circ \]

### Step 3: Opposite Angles are Equal
Opposite angles formed by the intersection of two lines are equal:

\[ \angle AOD = \angle BOC \quad \text{and} \quad \angle AOC = \angle BOD \]

### Step 4: Express Angles in Terms of x
From the given information:

- \( \angle AOD = 5x \)
- \( \angle AOC = 4x \)
- \( \angle BOC = 5x \) (since \( \angle AOD = \angle BOC \))
- \( \angle BOD = 4x \) (since \( \angle AOC = \angle BOD \))

### Step 5: Calculate the Value of x
Substituting the values into the sum equation:

\[ 5x + 4x + 5x + 4x = 360^\circ \]

Simplifying:

\[ 18x = 360^\circ \]

\[ x = 20^\circ \]

### Step 6: Calculate the Specific Angles
- \( \angle AOD = 5x = 5 \times 20^\circ = 100^\circ \)
- \( \angle AOC = 4x = 4 \times 20^\circ = 80^\circ \)
- \( \angle BOC = 5x = 100^\circ \)
- \( \angle BOD = 4x = 80^\circ \)

### Final Answer
- \( \angle AOD = 100^\circ \)
- \( \angle AOX = \angle AOD = 100^\circ \) (since X is on line AB)
- \( \angle BOD = 80^\circ \)
- \( \angle BOC = 100^\circ \)

Do you want more details or have any questions?

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Here are some related questions to deepen your understanding:

1. If \( \angle AOD = 100^\circ \), what is \( \angle COX \) when X lies on line CD?
2. How do the properties of intersecting lines help in calculating angles?
3. What would happen to the angles if the ratio \( \angle AOD : \angle AOC \) changed to 3:2?
4. Can you calculate the angles formed if AB and CD were perpendicular lines?
5. How does the sum of opposite angles help in verifying the correctness of the calculated angles?

**Tip:** Always remember that the sum of angles around a point is \( 360^\circ \), and opposite angles formed by two intersecting lines are equal.

Learn how to calculate angles AOD, AOX, BOD, and BOC in an intersecting lines problem. Understand the ratio of angles AOD to AOC and apply angle relationships around a point. Suitable for students in Grades 7-9.

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