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Let's solve Question 18.

**Given:**
- ABCD is a rectangle.
- Diagonals AC and BD intersect each other at point O.
- \(\angle AOD = 64^\circ\).

**To Find:**
- The values of \(p\) and \(q\).

**Solution:**

1. **Understanding the Properties of a Rectangle:**
   - In a rectangle, the diagonals bisect each other.
   - The diagonals of a rectangle are equal in length.
   - Each diagonal divides the rectangle into two congruent right-angled triangles.

2. **Angle Relationships:**
   Since ABCD is a rectangle, diagonals \(AC\) and \(BD\) bisect each other at point \(O\). Therefore, \(\triangle AOD\), \(\triangle BOC\), \(\triangle AOB\), and \(\triangle COD\) are congruent to each other.

3. **Given Angle:**
   \(\angle AOD = 64^\circ\).

4. **Calculate Other Angles:**
   In a rectangle, diagonals bisect each other and form four angles at the intersection point \(O\). These four angles are equal.

   Therefore,
   \[
   \angle AOD = \angle BOC = \angle AOB = \angle COD = 64^\circ.
   \]

5. **Finding \(p\) and \(q\):**

   Let:
   - \(p = \angle AOD\)
   - \(q = \angle AOB\)

   Since \(\angle AOD = 64^\circ\) and \(\angle AOB = 64^\circ\), we have:
   \[
   p = 64^\circ, \quad q = 64^\circ.
   \]

Thus, the values of \(p\) and \(q\) are both \(64^\circ\).

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Would you like more details, or do you have any questions? Here are some related questions:

1. How do we know the diagonals of a rectangle bisect each other?
2. Why are the angles at the intersection of the diagonals equal?
3. What properties differentiate a rectangle from other parallelograms?
4. How would the angle values change if the shape was a rhombus instead of a rectangle?
5. Can a rectangle have diagonals that do not bisect at right angles?

**Tip:** Remember that in any quadrilateral, if the diagonals bisect each other and are equal, the shape is a rectangle.

In the given figure, ABCD is a rectangle. AC and BD intersect each other at O. If ∠AOD = 64°, then find p and q.

Solve a geometry problem involving a rectangle where diagonals intersect at an angle of 64°. Learn the properties of rectangles and how to determine the angles formed by intersecting diagonals. Suitable for Grade 8-10 students.

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