Student completing a geometry proof worksheet on parallel lines and transversals.

Let's go through both proofs step by step and explain the reasoning. The goal is to understand and complete the proof for each problem, especially filling in any missing parts.

---

🔷 Problem 1: Prove ∠1 ≅ ∠8 (Alternate Exterior Angles are Congruent)

Given: Lines $ g $ and $ h $ are parallel lines cut by a transversal.

Prove: $ \angle 1 \cong \angle 8 $

| Statements | Reasons |
|----------------|-------------|
| 1. Lines $ g $ and $ h $ are parallel lines cut by a transversal | 1. Given |
| 2. $ \angle 1 \cong \angle 4 $ | 2. Vertical angles |
| 3. $ \angle 4 \cong \angle 8 $ | 3. Corresponding angles |
| 4. $ \angle 1 \cong \angle 8 $ | 4. Substitution |

✔ Explanation:

- Step 2: $ \angle 1 $ and $ \angle 4 $ are vertical angles — they are opposite angles formed by two intersecting lines. Vertical angles are always congruent.
- Step 3: $ \angle 4 $ and $ \angle 8 $ are corresponding angles. Since $ g \parallel h $, corresponding angles are congruent.
- Step 4: Since $ \angle 1 \cong \angle 4 $ and $ \angle 4 \cong \angle 8 $, by substitution, $ \angle 1 \cong \angle 8 $. This proves that alternate exterior angles are congruent when lines are parallel.

✔️ This proof correctly uses the transitive property of congruence via substitution.

---

🔷 Problem 2: Prove ∠2 ≅ ∠6 (Corresponding Angles are Congruent)

Wait — actually, the statement says "Prove: ∠2 ≅ ∠6", but it says "corresponding angles are congruent" as the reason. Let’s check if this is correct.

But first, let's analyze the diagram:

From the image:
- Line $ g $ and $ h $ are parallel.
- A transversal cuts them.
- Angles are labeled 1–8 around the intersection points.

Standard labeling:
- At the top line ($ g $): ∠1, ∠2, ∠3, ∠4 (going clockwise)
- At the bottom line ($ h $): ∠5, ∠6, ∠7, ∠8

So:
- ∠1 and ∠5 are corresponding
- ∠2 and ∠6 are corresponding
- ∠3 and ∠7 are corresponding
- ∠4 and ∠8 are corresponding

Thus, ∠2 and ∠6 are corresponding angles.

But the proof says:
> Prove: $ \angle 2 \cong \angle 6 $ (corresponding angles are congruent)

That seems circular — we're proving that corresponding angles are congruent using the fact that corresponding angles are congruent?

Wait — actually, this proof is not proving the general rule, but rather using properties to show $ \angle 2 \cong \angle 6 $, possibly assuming some known theorems.

Let’s look at the proof:

| Statements | Reasons |
|----------------|-------------|
| 1. Lines $ g $ and $ h $ are parallel lines cut by a transversal | 1. Given |
| 2. $ \angle 2 \cong \angle 7 $ | 2. Alternate exterior angles |
| 3. $ \angle 6 \cong \angle 7 $ | 3. Vertical angles |
| 4. $ \angle 2 \cong \angle 6 $ | 4. ? |

We need to fill in Reason 4.

---

#### ✔ Step-by-step Explanation:

- Step 2: $ \angle 2 $ and $ \angle 7 $ are alternate exterior angles. Since $ g \parallel h $, alternate exterior angles are congruent → So this is valid.
- Step 3: $ \angle 6 $ and $ \angle 7 $ are vertical angles → vertical angles are congruent → so $ \angle 6 \cong \angle 7 $
- Step 4: We now have:
- $ \angle 2 \cong \angle 7 $
- $ \angle 6 \cong \angle 7 $
→ So $ \angle 2 \cong \angle 6 $ by transitive property of congruence.

So, Reason 4 should be:
👉 Transitive Property of Congruence

---

✔ Final Answer:

For Problem 2, the missing reason in Step 4 is:

> 4. Transitive Property of Congruence

---

📌 Summary:

- Problem 1 correctly proves that alternate exterior angles are congruent using vertical angles and corresponding angles.
- Problem 2 uses alternate exterior angles and vertical angles to prove that corresponding angles are congruent, which is a common method to derive the corresponding angles postulate from other angle relationships.

Even though the statement says “Prove: ∠2 ≅ ∠6 (corresponding angles are congruent)”, it’s likely meant to demonstrate that corresponding angles are congruent using other known facts (like alternate exterior and vertical angles), so the proof is valid.

---

✔ Final Boxed Answer:

> The missing reason in Problem 2, Step 4, is: *Transitive Property of Congruence*.