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 $64^\circ$.

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.