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It looks like you've uploaded a geometry proof that you're working on. The problem asks you to complete a proof of Theorem 51 with the following:

### Given:
- \(\bigcirc O\)
- \(\overline{AB} \perp \overline{HI}\)
- \(\overline{CD} \perp \overline{GJ}\)
- \(OE \cong OF\)

### To Prove:
\[
AB \cong CD
\]

The proof consists of a list of statements with missing reasons. Let’s go through the missing justifications for each step of the proof:

### Steps of the Proof (with Reasons):

1. **Given**
   - Already filled in.
   
2. **\(OE \cong OF\)**
   - This is a given.

3. **Draw radii \(OA\) and \(OC\)**
   - Reason: **Two points determine a straight line**.

4. **\(OA \cong OC\)** 
   - Reason: **All radii of a circle are congruent**.

5. **\(\angle AEO\) and \(\angle CFO\) are right angles**
   - Reason: **Perpendicular lines form right angles** (from the fact that \(AB \perp HI\) and \(CD \perp GJ\)).

6. **\(\Delta AEO\) and \(\Delta CFO\) are right triangles**
   - Reason: **Triangles with right angles are right triangles**.

7. **\(\Delta AEO \cong \Delta CFO\)**
   - Reason: **Hypotenuse-leg (HL) congruence theorem** (since \(OE \cong OF\), \(OA \cong OC\), and \(\angle AEO\) and \(\angle CFO\) are right angles).

8. **\(AE \cong CF\)**
   - Reason: **Corresponding parts of congruent triangles (CPCTC)**.

9. **\(AE = CF\)**
   - Reason: **Definition of congruence** (congruent segments have equal lengths).

10. **\(2AE = 2CF\)**
    - Reason: **Multiplication property** (both sides are multiplied by 2).

11. **Diameter \(HI\) bisects \(\overline{AB}\); diameter \(GJ\) bisects \(\overline{CD}\)**
    - Reason: **A diameter bisects a chord perpendicular to it** (a theorem about diameters and chords in a circle).

12. **\(AE = EB\); \(CF = FD\)**
    - Reason: **Definition of betweenness of points** (the diameter bisects the chords).

13. **\(AB = AE + EB\); \(CD = CF + FD\)**
    - Reason: **Segment addition postulate**.

14. **\(AB = AE + AE; CD = CF + CF\)**
    - Reason: **Substitution** (since \(EB = AE\) and \(FD = CF\)).

15. **\(AB = 2AE; CD = 2CF\)**
    - Reason: **Addition**.

16. **\(AB = CD\)**
    - Reason: **Substitution** (from step 10, \(2AE = 2CF\)).

17. **Thus, \(AB \cong CD\)**
    - Reason: **Definition of congruent segments** (segments with equal lengths are congruent).

Let me know if you'd like more details on any step, or if you have further questions! 

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Here are some related questions that might help you expand your understanding:

1. What is the Hypotenuse-Leg (HL) theorem, and when is it applicable?
2. How does the property of a diameter bisecting a chord apply to this proof?
3. Why is it important to identify corresponding parts in congruent triangles?
4. Can you apply the segment addition postulate in other geometric problems?
5. How do perpendicular lines ensure right angles in this context?

**Tip:** Always label and identify congruent triangles early in a proof, as they often provide a direct path to proving other segments or angles congruent through CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Complete this proof of Theorem 51. Given: Circle O; AB perpendicular to HI, CD perpendicular to GJ; OE congruent to OF. Prove: AB congruent to CD.

Complete the proof of Theorem 51, demonstrating that AB is congruent to CD in a circle. This step-by-step solution applies the Hypotenuse-Leg Congruence Theorem, CPCTC, and other geometry principles. Suitable for students in Grades 9-11.

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