Proof that angles 7 and 6 are supplementary when two parallel lines are intersected by a transversal.

Let's solve this geometry proof step by step and fill in the missing parts of the table.

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Given:

- $ r \parallel q $
- Lines $ r $ and $ q $ are intersected by transversal $ p $

To Prove:

- $ \angle 7 $ and $ \angle 6 $ are supplementary (i.e., $ m\angle 7 + m\angle 6 = 180^\circ $)

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We are given a two-column proof with some statements and reasons. Let's complete it logically.

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Step-by-step Analysis:

#### Statement 1:
> $ r \parallel q $; $ r $ and $ q $ are intersected by transversal $ p $

Reason: Given ✔

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#### Statement 2:
> $ m\angle 8 \cong m\angle 6 $

This means angle 8 is congruent to angle 6.

Let’s look at the diagram:

- $ \angle 8 $ and $ \angle 6 $ are corresponding angles because:
- They are on the same side of the transversal $ p $
- One is above line $ r $, the other above line $ q $
- Since $ r \parallel q $, corresponding angles are congruent.

So,

Reason 2: Corresponding Angles Postulate
(When two parallel lines are cut by a transversal, corresponding angles are congruent.)

✔ So, Reason 2 is: Corresponding Angles Postulate

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#### Statement 3:
> $ \angle 8 $ and $ \angle 7 $ are supplementary

Why? Because $ \angle 8 $ and $ \angle 7 $ form a linear pair — they are adjacent angles that lie on a straight line (line $ r $), so their measures add up to $ 180^\circ $.

So,

Reason 3: Definition of supplementary angles
(Because they form a linear pair and are adjacent on a straight line.)

But the blank says: "definition of" → we need to complete it.

Actually, the reason should be: Linear Pair Postulate, or more precisely, definition of supplementary angles (since supplementary angles sum to 180°, and these two form a straight line).

But since the instruction says "definition of", likely they want:

"definition of supplementary angles" — but wait, that's not quite right.

Wait — actually, "supplementary angles" are defined as two angles whose measures add to 180°. But here, we're saying they are supplementary because they form a linear pair.

So better reasoning: Linear Pair Postulate — which states that if two angles form a linear pair, then they are supplementary.

But the blank says: "definition of"

Hmm. Maybe they expect: "definition of a linear pair"?

No — the definition of a linear pair is two adjacent angles forming a straight line.

But the statement says: "$ \angle 8 $ and $ \angle 7 $ are supplementary" — so the reason is that they form a linear pair, hence supplementary.

So the correct reason is: Linear Pair Postulate

But the table says: "definition of" — so perhaps it's expecting: "definition of supplementary angles"?

Wait — no. That would be circular. We’re proving they are supplementary, so we can't use that as a reason.

Better approach: The reason for why $ \angle 8 $ and $ \angle 7 $ are supplementary is that they form a linear pair.

So the correct reason is: Linear Pair Postulate

But the blank says: "definition of" — so maybe it's expecting: "definition of a linear pair"?

Actually, the definition of a linear pair is: two adjacent angles whose non-common sides form a straight line.

But the conclusion is that they are supplementary — that comes from the postulate, not just the definition.

However, in many textbooks, they say:

> "Angles that form a linear pair are supplementary" — this is a postulate.

So perhaps the intended answer is: "linear pair" — meaning: "definition of a linear pair" leads to supplementary angles?

But that's not quite accurate.

Alternatively, maybe the blank is meant to be filled as: "linear pair", and the full reason is: "definition of linear pair" → but that doesn't directly say they are supplementary.

Wait — let's think again.

The statement is: $ \angle 8 $ and $ \angle 7 $ are supplementary.

The reason must justify that.

They are adjacent and form a straight line → so they are a linear pair → therefore, supplementary.

So the best reason is: Linear Pair Postulate

But the blank says: "definition of" — so perhaps it's a typo or misphrasing.

Alternatively, maybe the blank is meant to be completed as: "linear pair", and the full reason is: "definition of linear pair" — but that’s not sufficient.

Wait — actually, the definition of supplementary angles is that their measures add to 180°.

But we don’t know that yet — we’re trying to show it.

So we can't use that.

Therefore, the correct reason is Linear Pair Postulate, even if the table says "definition of".

But looking at the structure, perhaps the intended answer is:

"definition of supplementary angles" — but that’s backward.

Wait — no.

Alternatively, maybe the blank is meant to be: "straight angles" or something?

No.

Let me check standard proofs.

In most geometry proofs, when two angles form a linear pair, the reason is:

> Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.

So the reason should be: Linear Pair Postulate

But the table says: "definition of" — so maybe it's expecting: "supplementary angles"?

That doesn’t make sense.

Wait — perhaps the blank is for "supplementary angles", and the full reason is: "definition of supplementary angles" — but again, we’re trying to prove they are supplementary, so we can't use that.

I think there might be a mistake in the table setup.

But let's look at the next step.

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#### Statement 4:
> $ \angle 7 $ and $ \angle 6 $ are supplementary

Because we know:

- $ \angle 8 \cong \angle 6 $ (from Statement 2)
- $ \angle 8 $ and $ \angle 7 $ are supplementary (Statement 3)
- So, substitute $ \angle 6 $ for $ \angle 8 $ → $ \angle 6 $ and $ \angle 7 $ are supplementary

So, Statement 4 should be:
> $ \angle 7 $ and $ \angle 6 $ are supplementary

And Reason 4: substitution

But we need to write it properly.

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Now, let’s fill in the table:

| Statement | Reason |
|----------|--------|
| 1. $ r \parallel q $; $ r $ and $ q $ are intersected by transversal $ p $ | 1. Given |
| 2. $ m\angle 8 \cong m\angle 6 $ | 2. Corresponding Angles Postulate |
| 3. $ \angle 8 $ and $ \angle 7 $ are supplementary | 3. Linear Pair Postulate |
| 4. $ \angle 7 $ and $ \angle 6 $ are supplementary | 4. Substitution |

But the table says for reason 3: "definition of" — so what could it be?

Wait — perhaps they consider "linear pair" as the definition.

But the correct term is Linear Pair Postulate.

Alternatively, maybe the blank is meant to be: "linear pair", and the reason is: "definition of linear pair" — but that's not enough.

Wait — perhaps in this context, they define supplementary angles via linear pairs.

But standardly, the reason is Linear Pair Postulate.

However, since the blank says "definition of", maybe they want: "supplementary angles"?

No — again, we're proving they are supplementary.

Wait — here’s a possibility: the definition of supplementary angles is that two angles are supplementary if their measures add to 180°.

But we are not using that definition to conclude they are supplementary — we are using the fact that they form a linear pair.

So I think the table has a slight error.

But let’s assume that the intended answer is:

> Reason 3: definition of supplementary angles — but that’s incorrect.

Wait — unless they mean: "definition of a linear pair"?

But the statement is about being supplementary.

Alternatively, maybe the blank is meant to be: "straight angle"?

No.

Another idea: $ \angle 8 $ and $ \angle 7 $ form a straight angle (180°), so they are supplementary.

So the reason could be: "definition of a straight angle"?

Yes! That might be it.

- A straight angle measures 180°.
- $ \angle 8 $ and $ \angle 7 $ together form a straight angle along line $ r $.
- So their sum is 180° → they are supplementary.

So Reason 3 could be: "definition of a straight angle"

That makes sense.

Because:
- The two angles form a straight line → they form a straight angle → measure 180° → supplementary.

So:

Reason 3: Definition of a straight angle

✔ That fits.

Now, for Statement 4:

We have:
- $ m\angle 8 = m\angle 6 $ (from Step 2)
- $ m\angle 8 + m\angle 7 = 180^\circ $ (from Step 3)
- Substitute $ m\angle 6 $ for $ m\angle 8 $ → $ m\angle 6 + m\angle 7 = 180^\circ $
- So $ \angle 6 $ and $ \angle 7 $ are supplementary

So:

Statement 4: $ \angle 7 $ and $ \angle 6 $ are supplementary
Reason 4: Substitution

Perfect.

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Final Completed Table:

| Statement | Reason |
|----------|--------|
| 1. $ r \parallel q $; $ r $ and $ q $ are intersected by transversal $ p $ | 1. Given |
| 2. $ m\angle 8 \cong m\angle 6 $ | 2. Corresponding Angles Postulate |
| 3. $ \angle 8 $ and $ \angle 7 $ are supplementary | 3. Definition of a straight angle |
| 4. $ \angle 7 $ and $ \angle 6 $ are supplementary | 4. Substitution |

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✔ Explanation:

- Since $ r \parallel q $, and $ p $ is a transversal, $ \angle 8 $ and $ \angle 6 $ are corresponding angles, so they are congruent.
- $ \angle 8 $ and $ \angle 7 $ are adjacent and form a straight line (on line $ r $), so they form a straight angle, which measures 180° — thus, they are supplementary.
- By substitution, since $ \angle 8 \cong \angle 6 $, replacing $ \angle 8 $ with $ \angle 6 $ in the supplementary pair gives that $ \angle 6 $ and $ \angle 7 $ are also supplementary.

Hence, $ \angle 7 $ and $ \angle 6 $ are supplementary.

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✔ Answer:

Completed Proof Table:

| Statement | Reason |
|----------|--------|
| 1. $ r \parallel q $; $ r $ and $ q $ are intersected by transversal $ p $ | 1. Given |
| 2. $ m\angle 8 \cong m\angle 6 $ | 2. Corresponding Angles Postulate |
| 3. $ \angle 8 $ and $ \angle 7 $ are supplementary | 3. Definition of a straight angle |
| 4. $ \angle 7 $ and $ \angle 6 $ are supplementary | 4. Substitution |

This proves that $ \angle 7 $ and $ \angle 6 $ are supplementary.