To solve this problem, we first need to analyze the properties of the rhombus and the relationship between the two equilateral triangles and the isosceles triangles.

Step 1: Properties of a Rhombus

A rhombus is a quadrilateral where all four sides are equal. The diagonals of a rhombus bisect each other at right angles (90°). Since the rhombus in question is formed by joining two equilateral triangles, we can conclude that the rhombus is symmetric.

Step 2: Properties of the Equilateral Triangles

In an equilateral triangle, all three angles are equal to $60^\circ$, and all sides are equal in length. Two of these triangles form the rhombus, so their combined interior angles at the common vertex must be:

$60^\circ + 60^\circ = 120^\circ$

Thus, one of the angles in the rhombus is $120^\circ$, and since opposite angles in a rhombus are equal, the other angle at the opposite vertex is also $120^\circ$.

Step 3: Other Angles of the Rhombus

Since the sum of all interior angles in any quadrilateral is $360^\circ$, and we already know that two of the angles in the rhombus are $120^\circ$ each, the other two angles must sum up to $360^\circ - (120^\circ + 120^\circ) = 120^\circ$.

Since these two angles are equal (opposite angles in a rhombus are equal), each of them must be:

$\frac{120^\circ}{2} = 60^\circ$

Step 4: The Isosceles Triangle

The rhombus can also be formed by joining two isosceles triangles. We know that the angle between the two equal sides of the rhombus is $120^\circ$ (from the equilateral triangles). This angle must correspond to the vertex angle of the isosceles triangle.

The sum of the angles in any triangle is $180^\circ$. Therefore, the sum of the two base angles in the isosceles triangle is:

$180^\circ - 120^\circ = 60^\circ$

Since the base angles are equal, each base angle of the isosceles triangle is:

$\frac{60^\circ}{2} = 30^\circ$

Conclusion:

Each isosceles triangle has:

• One vertex angle of $120^\circ$,
• Two base angles of $30^\circ$ each.

Would you like further clarification on any step or details?

Related Questions:

1. How do the properties of a rhombus influence the angles of the triangles forming it?
2. What are the key differences between equilateral and isosceles triangles?
3. How would the solution change if the rhombus was made using non-equilateral triangles?
4. Can a rhombus always be split into two congruent isosceles triangles?
5. How do the diagonals of a rhombus affect its internal angles?

Tip:

Always remember that the sum of the angles in any triangle is $180^\circ$, and this fundamental property is key to solving many geometric problems!