Prove that vertically opposite angles are equal.

Answer

To prove that vertically opposite angles are equal, we will follow a structured approach. Let’s denote the intersecting lines and angles clearly.

### Step-by-Step Solution:

1. **Identify the Intersecting Lines and Angles:**
- Let lines A and B intersect at point O.
- This creates four angles: ∠AOB, ∠BOC, ∠COD, and ∠DOA.
- We need to prove that ∠AOD = ∠BOC (these are vertically opposite angles).

2. **Use the Linear Pair Property:**
- Notice that ∠AOB and ∠AOD form a linear pair. Therefore, the sum of these angles is 180 degrees.
- This can be written as:
\[ \angle AOB + \angle AOD = 180^\circ \quad \text{(Equation 1)} \]

3. **Identify Another Linear Pair:**
- Similarly, ∠BOC and ∠AOD also form a linear pair. Therefore, their sum is also 180 degrees:
\[ \angle BOC + \angle AOD = 180^\circ \quad \text{(Equation 2)} \]

4. **Set Up the Equations:**
- From Equation 1:
\[ \angle AOB + \angle AOD = 180^\circ \]
- From Equation 2:
\[ \angle BOC + \angle AOD = 180^\circ \]

5. **Subtract the Equations:**
- Now, we can subtract Equation 1 from Equation 2:
\[ (\angle BOC + \angle AOD) - (\angle AOB + \angle AOD) = 180^\circ - 180^\circ \]
- This simplifies to:
\[ \angle BOC - \angle AOB = 0 \]
- Thus, we have:
\[ \angle BOC = \angle AOB \]

6. **Conclude the Proof:**
- Since we have established that ∠AOD = ∠BOC, we can conclude that:
\[ \angle AOD = \angle BOC \]
- Therefore, vertically opposite angles are equal.

### Final Statement:
Hence, we have proved that vertically opposite angles are equal.

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