Proofs in Geometry: Complete with ease | airSlate SignNow - Free Printable

It looks like you've uploaded a worksheet titled "Worksheet 2.6 Geometric Proof" with two geometry problems involving angle relationships and proofs. Since I can't view images directly, I'll help you solve the problems based on the text you provided.

Let’s go through each problem step by step.

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Problem 1:

Given:
- ∠1 and ∠2 form a linear pair.
- ∠3 and ∠4 form a linear pair.
Prove:
m∠1 + m∠2 + m∠3 + m∠4 = 360°

#### Step-by-step proof:

We are to fill in the blanks of a two-column proof.

| Statements | Reasons |
|----------------|-------------|
| 1. ∠1 and ∠2 form a linear pair; ∠3 and ∠4 form a linear pair | 1. Given |
| 2. ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary | 2. Definition of linear pair (linear pairs are supplementary) |
| 3. m∠1 + m∠2 = 180°; m∠3 + m∠4 = 180° | 3. Definition of supplementary angles |
| 4. m∠1 + m∠2 + m∠3 + m∠4 = 360° | 4. Addition property of equality (add equations from step 3) |

✔ Explanation:
A linear pair consists of two adjacent angles whose non-common sides form a straight line. By definition, such angles are supplementary (sum to 180°). Since both pairs (∠1 & ∠2) and (∠3 & ∠4) are linear pairs, each sums to 180°. Adding them together gives 180° + 180° = 360°.

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Problem 2:

Given:
m∠2 = m∠3 = m∠4 = 180°
Wait — this seems incorrect.

Hold on — if m∠2 = m∠3 = m∠4 = 180°, that would mean each of those angles is a straight angle. But in a triangle, angles must be less than 180°, and their sum is 180°.

So likely there's a typo in your transcription.

Let me assume the correct version is:

> Given: m∠1 + m∠2 + m∠3 = 180°
> Prove: m∠1 : m∠2 : m∠3 = 1 : 2 : 4

But that doesn’t make sense unless more info is given.

Alternatively, perhaps the given is:

> m∠1 = x, m∠2 = 2x, m∠3 = 4x
> Prove: m∠1 + m∠2 + m∠3 = 180° → so x + 2x + 4x = 7x = 180° → x ≈ 25.7°

But the problem says:
Given: m∠2 = m∠3 = m∠4 = 180° → This can't be right.

Wait — looking at the diagram (which I can't see), it appears to be a triangle with an exterior angle labeled ∠4.

Possibly, the correct given is:

> m∠1 + m∠2 + m∠3 = 180° (triangle angle sum)
> and ∠4 is an exterior angle to the triangle, formed by extending one side.

And we're told:
> m∠2 = m∠3 = m∠4 = 180°? No — that’s impossible.

Let’s reconsider.

Maybe the correct given is:

> m∠1 = x, m∠2 = 2x, m∠3 = 4x
> and we need to prove that these three angles form a triangle (i.e., sum to 180°)

Then:

| Statements | Reasons |
|----------------|-------------|
| 1. m∠1 = x, m∠2 = 2x, m∠3 = 4x | 1. Given |
| 2. m∠1 + m∠2 + m∠3 = 180° | 2. Triangle Angle Sum Theorem |
| 3. x + 2x + 4x = 180° | 3. Substitution |
| 4. 7x = 180° | 4. Combine like terms |
| 5. x = 180°/7 ≈ 25.7° | 5. Division property of equality |

But the goal is to prove m∠1 : m∠2 : m∠3 = 1 : 2 : 4

Wait — actually, the problem says:

> Given: m∠2 = m∠3 = m∠4 = 180°
> Prove: m∠1 : m∠2 : m∠3 = 1 : 2 : 4

That still doesn't make sense.

Perhaps it's a typo and should be:

> Given: m∠1 : m∠2 : m∠3 = 1 : 2 : 4
> Prove: m∠1 + m∠2 + m∠3 = 180°

Or maybe:

> Given: ∠1, ∠2, ∠3 are angles of a triangle with ratio 1:2:4
> Prove: The measures are consistent with triangle angle sum.

Let’s assume the intended problem is:

> Given: In triangle ABC, ∠A : ∠B : ∠C = 1 : 2 : 4
> Prove: The angles sum to 180°

Let’s solve that.

Let:
- m∠A = x
- m∠B = 2x
- m∠C = 4x

Sum: x + 2x + 4x = 7x = 180°
→ x = 180° / 7 ≈ 25.71°

So the angles are approximately:
- ∠A = 25.7°
- ∠B = 51.4°
- ∠C = 102.9°

This is valid (all positive, sum to 180°).

So the proof would be:

| Statements | Reasons |
|----------------|-------------|
| 1. m∠1 : m∠2 : m∠3 = 1 : 2 : 4 | 1. Given |
| 2. Let m∠1 = x, m∠2 = 2x, m∠3 = 4x | 2. Ratio definition |
| 3. m∠1 + m∠2 + m∠3 = 180° | 3. Triangle Angle Sum Theorem |
| 4. x + 2x + 4x = 180° | 4. Substitution |
| 5. 7x = 180° | 5. Simplify |
| 6. x = 180° / 7 | 6. Division property |
| 7. So m∠1 = 180°/7, m∠2 = 360°/7, m∠3 = 720°/7 | 7. Substitution |
| 8. Therefore, m∠1 : m∠2 : m∠3 = 1 : 2 : 4 | 8. Ratio confirmed |

But wait — the problem says Prove: m∠1 : m∠2 : m∠3 = 1 : 2 : 4, but that’s given. So likely the given is different.

Alternatively, perhaps the given is that ∠4 is an exterior angle, and m∠4 = m∠1 + m∠2 (exterior angle theorem), and some other conditions.

Given the diagram shows a triangle with an exterior angle labeled ∠4, and angles inside labeled ∠1, ∠2, ∠3.

Let’s suppose:

> Given: ∠1 and ∠2 are interior angles of triangle, ∠4 is exterior angle at vertex C, so ∠4 = ∠1 + ∠2
> Also, m∠2 = m∠3, and m∠4 = 180°? No — that can't be.

Wait — perhaps the given is:

> m∠1 = 1/2 m∠2, m∠2 = m∠3, and m∠4 = 180° – m∠3 (since ∠3 and ∠4 form a linear pair)

But without seeing the image, it's hard.

Let me try to interpret based on standard problems.

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Most Likely Intended Problem 2:

> Given: In triangle ABC, ∠A = x, ∠B = 2x, ∠C = 4x
> Prove: x + 2x + 4x = 180°

But again, that’s just calculation.

Alternatively:

> Given: ∠1 and ∠2 are complementary, ∠3 and ∠4 are supplementary... no, not matching.

Wait — let’s look back.

You wrote:

> "2. Given: m∠2 = m∠3 = m∠4 = 180°
> Prove: m∠1 : m∠2 : m∠3 = 1 : 2 : 4"

This is impossible because no angle in a triangle can be 180°.

So likely, the given is:

> m∠1 = x, m∠2 = 2x, m∠3 = 4x
> and ∠4 is an exterior angle at ∠3, so ∠4 = ∠1 + ∠2

Then:
- m∠4 = x + 2x = 3x
- But also, since ∠3 and ∠4 form a linear pair: m∠3 + m∠4 = 180°
- So 4x + 3x = 180° → 7x = 180° → x = 180/7 ≈ 25.7°

Then:
- m∠1 = x = 180/7
- m∠2 = 2x = 360/7
- m∠3 = 4x = 720/7
- m∠4 = 3x = 540/7

Now, the ratio m∠1 : m∠2 : m∠3 = 1 : 2 : 4 — which matches.

So the goal is to prove that the ratio is 1:2:4, given the configuration.

But the given is probably:

> ∠1 and ∠2 are interior angles, ∠3 is another interior angle, ∠4 is exterior at ∠3, and m∠4 = m∠1 + m∠2
> and m∠3 = 4x, m∠2 = 2x, m∠1 = x
> or something similar.

Let’s write a plausible proof for Problem 2 based on standard geometry.

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Revised Problem 2 (Likely Intent):

> Given: In triangle ABC, ∠A = x, ∠B = 2x, ∠C = 4x
> Prove: The angles sum to 180°

But again, that’s not proving the ratio.

Wait — perhaps the given is:

> ∠1 and ∠2 are adjacent angles forming a straight line with ∠4, and ∠3 is vertical or something.

Without the image, it's ambiguous.

But based on the structure of the worksheet and common problems, here's a likely scenario:

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✔ Final Interpretation of Problem 2:

> Given:
> - ∠1, ∠2, and ∠3 are angles of a triangle.
> - ∠4 is an exterior angle at vertex C, so ∠4 = ∠1 + ∠2
> - ∠3 and ∠4 are supplementary (they form a linear pair)
> - m∠2 = 2 × m∠1, and m∠3 = 4 × m∠1

> Prove: m∠1 : m∠2 : m∠3 = 1 : 2 : 4

Let’s solve this.

Let m∠1 = x
Then m∠2 = 2x
m∠3 = 4x

From triangle angle sum:
m∠1 + m∠2 + m∠3 = 180°
x + 2x + 4x = 7x = 180°
x = 180° / 7

Now, ∠4 = m∠1 + m∠2 = x + 2x = 3x

Also, ∠3 and ∠4 are supplementary:
m∠3 + m∠4 = 180°
4x + 3x = 7x = 180° → same as above

So everything checks out.

Now, the ratio m∠1 : m∠2 : m∠3 = x : 2x : 4x = 1 : 2 : 4

So the proof would be:

| Statements | Reasons |
|----------------|-------------|
| 1. Let m∠1 = x | 1. Assume variable |
| 2. Then m∠2 = 2x, m∠3 = 4x | 2. Given ratios |
| 3. m∠1 + m∠2 + m∠3 = 180° | 3. Triangle Angle Sum Theorem |
| 4. x + 2x + 4x = 180° | 4. Substitution |
| 5. 7x = 180° | 5. Simplify |
| 6. x = 180° / 7 | 6. Solve |
| 7. So m∠1 = x, m∠2 = 2x, m∠3 = 4x | 7. Substitution |
| 8. Therefore, m∠1 : m∠2 : m∠3 = 1 : 2 : 4 | 8. Ratio definition |

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✔ Summary:

#### Problem 1:
- Use linear pair → supplementary angles → each pair sums to 180° → total 360°.

#### Problem 2:
- Assume angles in ratio 1:2:4
- Use triangle angle sum to confirm consistency
- Show ratio holds

If you can provide the exact text of the given and prove statements, or describe the diagram, I can give a precise solution.

But based on typical worksheets, the above is the most likely intended solution.

Let me know if you'd like me to format it neatly or draw the logic!