https://file.aiquickdraw.com/mathbot/file/1726079441138jsqrmxeh.jpg

Let's analyze the diagram and the questions based on it. The task asks to identify the postulate name, definition, or theorem justifying each statement. Here's how to approach them:

1. **Statement: If ∠1 ≅ ∠2, then BF is the bisector of ∠ABE.**
   - **Justification:** This is based on the **Angle Bisector Theorem**, which states that if a ray divides an angle into two congruent angles, it is an angle bisector. Since ∠1 ≅ ∠2, ray \( BF \) bisects ∠ABE.

2. **Statement: If \( BC \cong CG \), then C is the midpoint of \( \overline{BG} \).**
   - **Justification:** This follows from the **Definition of a Midpoint**, which states that the midpoint divides a segment into two equal (congruent) parts. Since \( BC \cong CG \), point C is the midpoint of \( \overline{BG} \).

3. **Statement: \( m\angle 3 + m\angle 4 = m\angle EBC \).**
   - **Justification:** This is a consequence of the **Angle Addition Postulate**, which states that if a point lies in the interior of an angle, the measure of the whole angle is the sum of the measures of the two smaller angles. Here, ∠EBC is formed by the sum of ∠3 and ∠4.

### Recap of Justifications:
1. **Angle Bisector Theorem**.
2. **Definition of a Midpoint**.
3. **Angle Addition Postulate**.

Would you like more detailed explanations of any of these theorems? 

Here are 5 related questions you can consider:
1. What is the definition of congruent angles?
2. How can you prove that two segments are congruent?
3. What other properties are associated with angle bisectors?
4. Can an angle have more than one bisector?
5. What is the relationship between complementary and supplementary angles?

**Tip:** Always ensure to label all parts of a diagram clearly when identifying bisectors or midpoints for easier reference!

Identify the postulate name, definition, or theorem that justifies each statement using a geometric diagram.

Learn how to identify geometric theorems and definitions that justify statements related to angle bisectors, midpoints, and angle addition. Suitable for students in Grades 8-10, this example covers key geometry concepts like the Angle Bisector Theorem, Definition of a Midpoint, and Angle Addition Postulate.

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