The angles made by the diagonals of a rhombus with one of its sides are in a 
4
:
5
4:5 ratio. Find the smaller angle of the rhombus.

Let's denote the angles made by the diagonals of the rhombus with one of its sides as \( 4x \) and \( 5x \), where \( x \) is a common factor. 

### Step 1: Relationship Between the Angles
Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four right-angled triangles. In each of these triangles, the angles adjacent to the sides of the rhombus sum up to \( 90^\circ \). Therefore, we have:

\[
4x + 5x = 90^\circ
\]

### Step 2: Solve for \( x \)
\[
9x = 90^\circ
\]
\[
x = 10^\circ
\]

### Step 3: Find the Angles
Now, substitute \( x \) back into the expressions for the angles:
- The first angle is \( 4x = 4 \times 10^\circ = 40^\circ \).
- The second angle is \( 5x = 5 \times 10^\circ = 50^\circ \).

### Step 4: Determine the Smaller Angle of the Rhombus
The smaller angle of the rhombus is twice the smaller angle made by the diagonals with the sides (since the diagonals bisect the angles at the vertices of the rhombus):

\[
\text{Smaller angle of the rhombus} = 2 \times 40^\circ = 80^\circ
\]

### Final Answer:
The smaller angle of the rhombus is \( \boxed{80^\circ} \).

Would you like more details on this solution or have any further questions?

Here are five related questions to consider:

1. How does the property of diagonals being perpendicular in a rhombus help in solving such problems?
2. What is the relationship between the angles and the sides in a rhombus?
3. How do you derive the formula for angles in a regular polygon like a rhombus?
4. Can you solve for the larger angle of the rhombus using similar methods?
5. What other geometric shapes have perpendicular diagonals, and how do their properties differ from those of a rhombus?

**Tip:** In any quadrilateral, especially in rhombuses, knowing the properties of the diagonals (like whether they bisect angles or are perpendicular) can greatly simplify angle calculations.

Learn how to find the smaller angle of a rhombus when the angles made by its diagonals with one of its sides are in a 4:5 ratio. Follow a step-by-step solution involving geometric properties and angle calculations. Suitable for high school students studying geometry.

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