Geometry proof worksheet demonstrating how to prove angle relationships using angle bisectors and vertical angles.

You're absolutely correct — the proof shown in the image is valid and logically complete. Let’s walk through it step by step to understand *why* it works.

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🎯 Goal: Prove that m∠2 = m∠4

Given:
> BF bisects ∠AFC

We need to use geometric definitions and properties to show that the measure of angle 2 equals the measure of angle 4.

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## ✔ Step-by-Step Breakdown:

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Statement 1:

BF bisects ∠AFC
→ *Reason: Given*

This is our starting point — we accept this as true from the problem statement.

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Statement 2:

∠1 ≅ ∠2
→ *Reason: Definition of an Angle Bisector*

✔ Explanation:
An angle bisector divides an angle into two congruent angles. Since BF bisects ∠AFC, it splits it into ∠1 and ∠2, which must be congruent.

So:
> ∠1 ≅ ∠2

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Statement 3:

∠1 ≅ ∠4
→ *Reason: Definition of Vertical Angles*

✔ Explanation:
Look at the diagram — ∠1 and ∠4 are vertical angles (they are opposite each other when two lines intersect — here, lines AD and BE intersect at F).

Vertical angles are always congruent by definition.

So:
> ∠1 ≅ ∠4

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Statement 4:

∠2 ≅ ∠4
→ *Reason: Substitution*

✔ Explanation:
From Statement 2: ∠1 ≅ ∠2
From Statement 3: ∠1 ≅ ∠4

Since both ∠2 and ∠4 are congruent to ∠1, by the Transitive Property of Congruence (or substitution), we can say:

> ∠2 ≅ ∠4

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Statement 5:

m∠2 = m∠4
→ *Reason: Definition of Congruent Angles*

✔ Explanation:
Congruent angles have equal measures. So if ∠2 ≅ ∠4, then their measures are equal.

> m∠2 = m∠4 ✔️

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## 🧠 Final Answer:

The proof is correct and follows valid geometric reasoning:

1. Use the given (angle bisector) → get ∠1 ≅ ∠2
2. Use vertical angles → get ∠1 ≅ ∠4
3. Substitute → ∠2 ≅ ∠4
4. Convert congruence to equality of measures → m∠2 = m∠4

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## ✍️ Bonus Tip:
This is a great example of combining multiple geometric concepts:
- Angle bisectors
- Vertical angles
- Transitive property / substitution
- Congruence vs. measure

You’ve nailed it! 👏

Let me know if you’d like to see a diagram or visualize the steps!