If two lines intersect each other, then the vertically opposite angles are equal.

Answer

To prove that if two lines intersect each other, then the vertically opposite angles are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Lines and Intersection Point:**
- Let the two lines be labeled as Line L and Line M.
- These lines intersect at a point O.

2. **Label the Angles:**
- When the lines intersect at point O, they form four angles. Let's label these angles as:
- Angle A (formed between Line L and Line M on one side)
- Angle B (adjacent to Angle A)
- Angle C (opposite to Angle A)
- Angle D (adjacent to Angle C)

3. **Understanding Vertically Opposite Angles:**
- Vertically opposite angles are the angles that are opposite each other when two lines intersect.
- In this case:
- Angle A is opposite to Angle C.
- Angle B is opposite to Angle D.

4. **Using the Property of Linear Pairs:**
- Angle A and Angle B are adjacent angles that form a linear pair. Therefore, they are supplementary:
- \( \text{Angle A} + \text{Angle B} = 180^\circ \) (1)
- Similarly, Angle C and Angle D are also adjacent angles that form a linear pair:
- \( \text{Angle C} + \text{Angle D} = 180^\circ \) (2)

5. **Setting Up the Equations:**
- From equation (1), we can express Angle B in terms of Angle A:
- \( \text{Angle B} = 180^\circ - \text{Angle A} \) (3)
- From equation (2), we can express Angle D in terms of Angle C:
- \( \text{Angle D} = 180^\circ - \text{Angle C} \) (4)

6. **Establishing Equality:**
- Since Angle B and Angle D are also adjacent angles, we can say:
- \( \text{Angle B} + \text{Angle D} = 180^\circ \) (5)
- Substituting equations (3) and (4) into equation (5):
- \( (180^\circ - \text{Angle A}) + (180^\circ - \text{Angle C}) = 180^\circ \)
- Simplifying this gives:
- \( 360^\circ - \text{Angle A} - \text{Angle C} = 180^\circ \)
- Thus, \( \text{Angle A} + \text{Angle C} = 180^\circ \)

7. **Conclusion:**
- Since Angle A and Angle C are equal and they are opposite angles, we can conclude that:
- \( \text{Angle A} = \text{Angle C} \)
- Similarly, we can show that:
- \( \text{Angle B} = \text{Angle D} \)
- Therefore, we have proved that vertically opposite angles are equal.

### Final Statement:
If two lines intersect each other, then the vertically opposite angles are equal.

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