Let's analyze this problem.

Observations:

1. $RSTV$ is a rhombus.

   • A rhombus has all sides equal, diagonals bisect each other at right angles, and the diagonals bisect the opposite angles.

2. The diagram shows:

   • $\triangle RT$ with a $36^\circ$ angle at $T$.
   • The diagonals $RV$ and $ST$ intersect at right angles.

3. The goal is to find the measures of angles:

   • $\angle 8, \angle 9, \angle 10, \angle 11,$ and $\angle 12$.

Solution:

Step 1: Angles from diagonals

• In a rhombus, diagonals bisect the angles.
  This means:
  • $\angle 8 = \angle 12$,
  • $\angle 9 = \angle 11$.

Step 2: Right-angle property of diagonals

• The diagonals intersect at $90^\circ$ (perpendicular).
  • In $\triangle RT$, since $\angle VTR = 90^\circ$ and $\angle RTV = 36^\circ$:
    • $\angle 10 = 90^\circ - 36^\circ = 54^\circ$.

Step 3: Using symmetry of diagonals

• $\triangle RTV$ and $\triangle SVT$ are congruent (diagonals divide the rhombus into four congruent right triangles).
  Therefore:
  • $\angle 9 = \angle 10 = 54^\circ$,
  • $\angle 8 = \angle 12 = 36^\circ$,
  • $\angle 11 = \angle 9 = 54^\circ$.

Final Answers:

1. $m\angle 8 = 36^\circ$
2. $m\angle 9 = 54^\circ$
3. $m\angle 10 = 54^\circ$
4. $m\angle 11 = 54^\circ$
5. $m\angle 12 = 36^\circ$

Would you like me to explain any specific part further?

Here are 5 related questions to deepen understanding:

1. Why are the diagonals of a rhombus always perpendicular?
2. How does the symmetry of a rhombus affect the properties of its angles?
3. What other properties can you deduce from the given information about $RSTV$?
4. Can you calculate the area of the rhombus if the diagonals are given?
5. What is the relationship between the diagonals and side lengths of a rhombus?

Tip: When solving problems about rhombuses, always use diagonal properties—they simplify calculations.