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We are given a rhombus where a straight line joins the midpoints of two sides, and we are asked to determine the size of angle \( a \).

### Key information:
- The rhombus has an interior angle of 80°, given in the diagram.
- The line joining the midpoints divides the rhombus into two trapezoids.

### Solution:
1. **Rhombus properties**: 
   - All sides of a rhombus are of equal length.
   - Opposite angles in a rhombus are equal.
   - Adjacent angles in a rhombus are supplementary (they add up to 180°).

2. **Using the angle of 80°**:
   - Since opposite angles are equal, the other angle at the opposite corner of the rhombus is also 80°.
   - The adjacent angles will then be supplementary, i.e., \( 180° - 80° = 100° \). Thus, each of the adjacent angles is 100°.

3. **Trapezoid created by the midpoint line**:
   - The line joining the midpoints of two sides of the rhombus is parallel to the other sides, and by symmetry, it divides the rhombus into two equal trapezoids.
   - This means the interior angles formed by this line will be split based on the geometry of the trapezoid.

4. **Angle \( a \)**:
   - Since the line joining the midpoints is parallel to one pair of opposite sides of the rhombus, angle \( a \) forms an alternate angle to the 80° angle in the rhombus.
   - By the property of parallel lines, alternate angles are equal. Therefore, \( a = 80° \).

Thus, the size of angle \( a \) is \( \boxed{80°} \).

Would you like more details or have any questions?

Here are five related questions to expand this topic:
1. How are the diagonals of a rhombus related to its angles?
2. What is the significance of the midpoint in geometric constructions?
3. Can you prove that the diagonals of a rhombus bisect each other at right angles?
4. How do trapezoid angle properties differ from those of a rhombus?
5. How would the solution change if the interior angle was different from 80°?

**Tip**: Always remember that parallel line properties, like alternate interior angles, can be useful for solving complex geometric problems.

Here is a rhombus. A straight line joining the midpoints of two of its sides is drawn. Work out the size of angle a.

Learn how to calculate the angle a in a rhombus where a straight line joins the midpoints of two sides. This problem covers geometric properties of rhombuses and uses the parallel line theorem to find alternate interior angles. Suitable for students in Grades 8-10.

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