State whether the statements are True or False.
Vertically opposite angles are equal.

Answer

To determine whether the statement "Vertically opposite angles are equal" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Vertically Opposite Angles**:
- When two lines intersect, they form four angles. The angles that are opposite each other are called vertically opposite angles.

2. **Labeling the Angles**:
- Let's label the intersection point of the two lines as point O. The angles formed can be labeled as follows:
- Angle AOB
- Angle BOC
- Angle COD
- Angle AOD

3. **Using the Linear Pair Property**:
- The angles that are adjacent to each other (next to each other) form a linear pair. The sum of angles in a linear pair is always 180 degrees.
- For example:
- Angle AOB + Angle BOC = 180 degrees (Equation 1)
- Angle AOD + Angle COD = 180 degrees (Equation 2)

4. **Setting Up the Equations**:
- From the above, we have:
- Equation 1: AOB + BOC = 180 degrees
- Equation 2: AOD + COD = 180 degrees

5. **Equating the Angles**:
- Since both equations equal 180 degrees, we can set the left-hand sides equal to each other:
- AOB + BOC = AOD + COD

6. **Rearranging the Equation**:
- Rearranging gives us:
- AOB - COD = BOC - AOD
- This implies that:
- AOB = COD (since BOC and AOD are equal)

7. **Conclusion**:
- Therefore, we have shown that vertically opposite angles AOB and COD are equal.
- Similarly, we can prove that angles AOD and BOC are also equal.
- Hence, the statement "Vertically opposite angles are equal" is **True**.

### Final Answer:
**True**

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