Given: Rhombus 
 with diagonal 


Prove: 
 bisects 
 and 



A diagram of a rhombus ABCD. A line AC is drawn as a diagonal.
Identify the missing statement and reason in the proof.

Given rhombus 
 with diagonal 
, it follows from the definition of a rhombus that 
. By the reflexive property of congruence, 
. So, 
 by the 
. Since corresponding parts of congruent triangles are congruent, 

 and 

. So, by the definition of segment bisector 
 bisects 
 and 
.



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).

---

### 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.

Given: Rhombus with diagonal. Prove: bisects and. A diagram of a rhombus ABCD. A line AC is drawn as a diagonal. Identify the missing statement and reason in the proof.

Explore the geometric proof that the diagonal of a rhombus bisects the angles, using the Side-Side-Side (SSS) postulate and congruent triangles. This detailed solution includes key geometry concepts and theorems, perfect for students in Grades 8-10.

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