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Solve. For exercises 16 to 18, make proof tables following the directions 

Let's address each of the exercises (16 to 18) as requested. From the image, the tasks involve proving statements in a geometric context. I'll create the proof tables according to the instructions.

### Exercise 16
- **Given:** \( \triangle \) ABC, \( AB \cong AC \)
- **To Prove:** \( \angle B \cong \angle C \)

| **Statements**                        | **Reasons**                                |
|---------------------------------------|--------------------------------------------|
| 1. \( AB \cong AC \)                  | Given                                      |
| 2. \( \angle A \) is common to both   | Reflexive Property                         |
| 3. \( \triangle ABC \cong \triangle ACB \) | SAS (Side-Angle-Side) Congruence          |
| 4. \( \angle B \cong \angle C \)      | Corresponding Parts of Congruent Triangles (CPCTC) |

### Exercise 17
- **Given:** \( \triangle ABC \) with \( AB \cong AC \)
- **To Prove:** \( AD \) bisects \( \angle BAC \)

| **Statements**                        | **Reasons**                                |
|---------------------------------------|--------------------------------------------|
| 1. \( AB \cong AC \)                  | Given                                      |
| 2. \( \angle ABD \cong \angle ACD \)  | Isosceles Triangle Theorem                 |
| 3. \( \angle BAD \cong \angle CAD \)  | Given that \( D \) lies on \( BC \), and both angles share a common side \( AD \) |
| 4. \( AD \) bisects \( \angle BAC \)  | Definition of an angle bisector            |

### Exercise 18
- **Given:** \( DE \cong DF \), \( \angle E \cong \angle F \)
- **To Prove:** \( D \) is the midpoint of \( EF \)

| **Statements**                        | **Reasons**                                |
|---------------------------------------|--------------------------------------------|
| 1. \( DE \cong DF \)                  | Given                                      |
| 2. \( \angle D \) is a common angle   | Reflexive Property                         |
| 3. \( \triangle DEF \) is isosceles   | Definition of Isosceles Triangle           |
| 4. \( D \) bisects \( EF \)           | Definition of a midpoint                   |

These proofs follow the typical steps to establish congruence or bisecting angles using given information and known geometric theorems.

Would you like to dive deeper into any of these exercises or do you have any additional questions?

Here are some related questions to consider:
1. How do we use the Side-Angle-Side (SAS) postulate to prove triangle congruence?
2. What is the significance of the Reflexive Property in geometric proofs?
3. Why does proving two angles congruent establish a bisector in isosceles triangles?
4. Can you identify other types of congruence postulates used in triangle proofs?
5. What would change in the proofs if the triangles were not isosceles?

**Tip:** When proving geometric statements, it's essential to use diagrams to visualize the problem. It helps in understanding the relationships between different parts of the figure.

Explore proof tables for exercises 16 to 18 involving triangle congruence and angle bisectors in geometric contexts. Learn how to prove statements using the Side-Angle-Side (SAS) Congruence Theorem and other relevant theorems. Suitable for high school students studying geometry.

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