The diagonals of a rectangle bisect each other

Answer

To prove that the diagonals of a rectangle bisect each other, we can follow these steps:

### Step 1: Draw the Rectangle
Draw a rectangle and label its vertices as \( A, B, C, D \) in clockwise order.

### Step 2: Draw the Diagonals
Draw the diagonals \( AC \) and \( BD \). Let the point where the diagonals intersect be \( O \).

### Step 3: Identify Angles
Since \( AB \) is parallel to \( CD \) and \( AD \) is parallel to \( BC \), we can identify pairs of alternate interior angles:
- Angle \( AOB \) is equal to angle \( COD \) (alternate interior angles).
- Angle \( AOD \) is equal to angle \( BOC \) (alternate interior angles).

### Step 4: Analyze Triangles
Now, consider triangles \( AOB \) and \( COD \):
- \( AO = OC \) (as we need to prove that diagonals bisect each other).
- \( BO = OD \) (as we need to prove that diagonals bisect each other).

### Step 5: Use Congruence Criteria
We can show that triangles \( AOB \) and \( COD \) are congruent using the Angle-Side-Angle (ASA) criterion:
- Angle \( AOB = Angle COD \) (from step 3).
- Side \( AO = OC \) (to be proved).
- Angle \( AOD = Angle BOC \) (from step 3).

### Step 6: Conclude the Proof
Since triangles \( AOB \) and \( COD \) are congruent, by the Corresponding Parts of Congruent Triangles (CPCT), we have:
- \( AO = OC \)
- \( BO = OD \)

This means that the diagonals \( AC \) and \( BD \) bisect each other at point \( O \).

### Conclusion
Thus, we have proved that the diagonals of a rectangle bisect each other.

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