Let's fill in the missing statements and reasons in the given proof for the rhombus.

Proof:

Given: Rhombus ABCD with diagonal AC.

1. It follows from the definition of a rhombus that $AB = AD$.

• Reason: Definition of a rhombus (all sides are congruent).

2. By the reflexive property of congruence, $AC = AC$.

• Reason: Reflexive property (any segment is congruent to itself).

3. So, $\triangle ABC \cong \triangle ADC$ by the Side-Side-Side (SSS) postulate.

• Reason: All three pairs of corresponding sides are congruent: $AB = AD$, $BC = CD$, and $AC = AC$.

4. Since corresponding parts of congruent triangles are congruent (CPCTC), $\angle BAC = \angle DAC$ and $\angle BCA = \angle DCA$.

• Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

5. So, by the definition of segment bisector, diagonal $AC$ bisects $\angle BAD$ and $\angle BCD$.

• Reason: Definition of angle bisector (diagonal divides the angles into two congruent angles).

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Completed Proof:

• Missing Statement: $\triangle ABC \cong \triangle ADC$
• Missing Reason: Side-Side-Side (SSS) postulate

Would you like more details on any part of the proof? Here are 5 relative questions to expand your understanding:

1. Why is the diagonal AC congruent to itself (reflexive property)?
2. How does the SSS postulate guarantee triangle congruence?
3. Can you prove that other properties of diagonals in a rhombus hold (like perpendicularity)?
4. What other properties of congruent triangles can we apply in rhombus proofs?
5. How does the definition of a rhombus help in proving other geometric theorems?

Tip: In a rhombus, the diagonals not only bisect the angles but are also perpendicular to each other, creating right triangles.