Geometry proof problems focusing on angle relationships and properties.

Let's solve each of these geometry proofs step by step, filling in the missing statements and reasons. These are all classic geometric proofs involving angle relationships.

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Problem 11:

Given: $ \overline{KM} $ bisects $ \angle JKL $
Prove: $ m\angle MKL = \frac{1}{2} m\angle JKL $

| Statements | Reasons |
|-----------|--------|
| 1. $ \overline{KM} $ bisects $ \angle JKL $ | 1. Given |
| 2. $ m\angle JKM = m\angle MKL $ | 2. Definition of Angle Bisector |
| 3. $ m\angle JKM + m\angle MKL = m\angle JKL $ | 3. Angle Addition Postulate |
| 4. $ m\angle MKL + m\angle MKL = m\angle JKL $ | 4. Substitution (from statement 2) |
| 5. $ 2m\angle MKL = m\angle JKL $ | 5. Simplify (combine like terms) |
| 6. $ m\angle MKL = \frac{1}{2} m\angle JKL $ | 6. Division Property of Equality |

✔ Explanation: Since $ KM $ bisects $ \angle JKL $, it splits the angle into two equal parts. Using substitution and algebraic manipulation, we show that each part is half the measure of the whole angle.

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Problem 12:

Given: $ \overline{BD} \perp \overline{BC} $, $ \angle ABD \cong \angle DBE $
Prove: $ \angle ABD $ and $ \angle EBC $ are complementary

| Statements | Reasons |
|-----------|--------|
| 1. $ \overline{BD} \perp \overline{BC} $ | 1. Given |
| 2. $ \angle DBC $ is a right angle | 2. Definition of Perpendicular Lines |
| 3. $ m\angle DBC = 90^\circ $ | 3. Definition of Right Angle |
| 4. $ m\angle DBE + m\angle EBC = m\angle DBC $ | 4. Angle Addition Postulate |
| 5. $ m\angle DBE + m\angle EBC = 90^\circ $ | 5. Substitution (from statement 3) |
| 6. $ \angle ABD \cong \angle DBE $ | 6. Given |
| 7. $ m\angle ABD = m\angle DBE $ | 7. Definition of Congruent Angles |
| 8. $ m\angle ABD + m\angle EBC = 90^\circ $ | 8. Substitution (from statement 7) |
| 9. $ \angle ABD $ and $ \angle EBC $ are complementary | 9. Definition of Complementary Angles (sum to 90°) |

✔ Explanation: Since $ BD \perp BC $, $ \angle DBC = 90^\circ $. The angles $ \angle DBE $ and $ \angle EBC $ make up this right angle. Because $ \angle ABD \cong \angle DBE $, replacing $ \angle DBE $ with $ \angle ABD $ gives us that $ \angle ABD $ and $ \angle EBC $ add to 90°, so they are complementary.

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Problem 13:

Given: $ \angle 1 $ and $ \angle 4 $ form a linear pair; $ \angle 1 $ and $ \angle 2 $ are supplementary
Prove: $ \angle 3 \cong \angle 4 $

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem (linear pairs are supplementary) |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 $ and $ \angle 1 $ are supplementary | 4. Same as above (just reordered) |
| 5. $ \angle 2 \cong \angle 4 $ | 5. Congruent Supplements Theorem (if two angles are supplementary to the same angle, they are congruent) |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property (since $ \angle 2 \cong \angle 4 $ and $ \angle 3 \cong \angle 2 $? Wait — need clarification.) |

Wait — there’s an issue here. Let's look at the diagram (implied):

From the image description:
- Two lines intersecting, forming angles 1, 2, 3, 4.
- $ \angle 1 $ and $ \angle 4 $ form a linear pair → adjacent and straight line → sum to 180°
- $ \angle 1 $ and $ \angle 2 $ are supplementary → also sum to 180°
- We are to prove $ \angle 3 \cong \angle 4 $

But from standard labeling:
- $ \angle 1 $ and $ \angle 2 $ are vertical angles? Or adjacent?
- Likely: $ \angle 1 $ and $ \angle 4 $ are adjacent (linear pair), so $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 $ and $ \angle 4 $ are both supplements of $ \angle 1 $ → $ \angle 2 \cong \angle 4 $ (Congruent Supplements Theorem)

Now, if $ \angle 3 $ and $ \angle 2 $ are vertical angles, then $ \angle 3 \cong \angle 2 $, so by transitivity $ \angle 3 \cong \angle 4 $

So let's fix the proof:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 $ and $ \angle 4 $ are supplementary to $ \angle 1 $ | 4. From statements 2 and 3 |
| 5. $ \angle 2 \cong \angle 4 $ | 5. Congruent Supplements Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property (assuming $ \angle 3 \cong \angle 2 $) — but wait, not stated yet! |

We're missing something. Let's re-evaluate.

Actually, looking at typical diagrams:
- When two lines intersect, opposite angles are vertical angles.
- If $ \angle 1 $ and $ \angle 4 $ form a linear pair → they are adjacent and on a straight line.
- Then $ \angle 3 $ is opposite $ \angle 1 $, and $ \angle 2 $ is opposite $ \angle 4 $?

Wait — more likely:
- $ \angle 1 $ and $ \angle 2 $ are adjacent (on one side)
- $ \angle 1 $ and $ \angle 4 $ are adjacent (on another side)
- But $ \angle 1 $ and $ \angle 4 $ are linear pair → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 \cong \angle 4 $ (same supplement of $ \angle 1 $)
- Also, $ \angle 3 $ and $ \angle 1 $ are vertical angles → $ \angle 3 \cong \angle 1 $? No, that doesn’t help.

Wait — perhaps the figure shows:

- $ \angle 1 $ and $ \angle 4 $ are adjacent and form a linear pair → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 \cong \angle 4 $ (by Congruent Supplements Theorem)
- Now, $ \angle 3 $ and $ \angle 2 $ are vertical angles → $ \angle 3 \cong \angle 2 $
- Therefore, $ \angle 3 \cong \angle 4 $ (Transitive)

But vertical angles are not given — so we must assume from diagram.

So correct proof:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 $ and $ \angle 4 $ are supplements of $ \angle 1 $ | 4. From 2 and 3 |
| 5. $ \angle 2 \cong \angle 4 $ | 5. Congruent Supplements Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Vertical Angles Theorem (if $ \angle 3 $ and $ \angle 2 $ are vertical) — but no!

Wait — actually, in most standard setups:
- $ \angle 1 $ and $ \angle 2 $ are adjacent
- $ \angle 1 $ and $ \angle 4 $ are adjacent
- $ \angle 3 $ is opposite $ \angle 1 $
- $ \angle 4 $ is opposite $ \angle 2 $

So $ \angle 3 $ and $ \angle 1 $ are vertical → $ \angle 3 \cong \angle 1 $
$ \angle 2 $ and $ \angle 4 $ are vertical → $ \angle 2 \cong \angle 4 $

But we have $ \angle 1 $ and $ \angle 2 $ supplementary → $ \angle 1 + \angle 2 = 180^\circ $
And $ \angle 1 $ and $ \angle 4 $ are linear pair → $ \angle 1 + \angle 4 = 180^\circ $

So $ \angle 2 \cong \angle 4 $ (supplements of same angle)

Then since $ \angle 3 \cong \angle 1 $ (vertical), and $ \angle 1 $ and $ \angle 4 $ are supplementary, etc.

But the goal is $ \angle 3 \cong \angle 4 $

This only happens if $ \angle 3 \cong \angle 4 $, which would require them to be vertical or congruent via other means.

But unless $ \angle 3 $ and $ \angle 4 $ are vertical, we can't say that.

Wait — perhaps the diagram shows:

- Two intersecting lines, forming four angles: $ \angle 1, \angle 2, \angle 3, \angle 4 $ around a point
- $ \angle 1 $ and $ \angle 4 $ are adjacent and form a linear pair → so they are on a straight line
- $ \angle 1 $ and $ \angle 2 $ are supplementary → so $ \angle 2 $ could be adjacent to $ \angle 1 $ on the other side

But in standard labeling, if $ \angle 1 $ and $ \angle 4 $ are a linear pair, then $ \angle 2 $ is opposite $ \angle 4 $, and $ \angle 3 $ is opposite $ \angle 1 $

So:
- $ \angle 1 $ and $ \angle 4 $ → adjacent, linear pair → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 = \angle 4 $ (both supplements of $ \angle 1 $)
- But $ \angle 2 $ and $ \angle 4 $ are vertical angles → $ \angle 2 \cong \angle 4 $
- $ \angle 3 $ and $ \angle 1 $ are vertical → $ \angle 3 \cong \angle 1 $
- But we want $ \angle 3 \cong \angle 4 $

So unless $ \angle 3 \cong \angle 4 $, which requires $ \angle 3 \cong \angle 1 $ and $ \angle 1 \cong \angle 4 $, which is only true if $ \angle 1 = 90^\circ $

So something’s off.

Wait — perhaps the diagram has a different configuration.

Alternatively, maybe the angles are labeled such that:

- $ \angle 1 $ and $ \angle 4 $ are a linear pair → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 = \angle 4 $ → $ \angle 2 \cong \angle 4 $
- Now, if $ \angle 3 $ and $ \angle 2 $ are vertical angles → $ \angle 3 \cong \angle 2 $
- Then $ \angle 3 \cong \angle 4 $ by transitive property

So likely, $ \angle 3 $ and $ \angle 2 $ are vertical angles.

Therefore, the proof should be:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 $ and $ \angle 4 $ are both supplementary to $ \angle 1 $ | 4. From 2 and 3 |
| 5. $ \angle 2 \cong \angle 4 $ | 5. Congruent Supplements Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property (only if $ \angle 3 \cong \angle 2 $) — but this is missing! |

Ah — we need to include that $ \angle 3 $ and $ \angle 2 $ are vertical angles.

But it's not stated.

Wait — maybe in the diagram, $ \angle 3 $ and $ \angle 2 $ are vertical, so we can state:

Step 5: $ \angle 3 \cong \angle 2 $ — Vertical Angles Theorem

Then Step 6: $ \angle 3 \cong \angle 4 $ — Transitive Property

But the table has only 6 steps, and step 6 says "Transitive Property", so likely:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 $ and $ \angle 4 $ are supplements of $ \angle 1 $ | 4. Definition of Supplementary Angles |
| 5. $ \angle 2 \cong \angle 4 $ | 5. Congruent Supplements Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property (but only if $ \angle 3 \cong \angle 2 $) |

But $ \angle 3 \cong \angle 2 $ is missing.

Unless $ \angle 3 $ and $ \angle 2 $ are vertical angles — which is implied by the diagram.

So perhaps the proof assumes that $ \angle 3 $ and $ \angle 2 $ are vertical, and thus congruent.

But it's not stated.

Alternatively, maybe the problem intends for us to use:

- $ \angle 1 $ and $ \angle 4 $ are supplementary → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 = \angle 4 $
- And $ \angle 3 $ and $ \angle 1 $ are vertical → $ \angle 3 = \angle 1 $
- But then $ \angle 3 $ and $ \angle 4 $ are not necessarily equal

So unless $ \angle 1 = \angle 4 $, which isn't given, this fails.

Wait — perhaps $ \angle 3 $ and $ \angle 4 $ are vertical angles?

That would make sense.

Maybe the labeling is:

- $ \angle 1 $ and $ \angle 4 $ are adjacent, forming a linear pair
- $ \angle 3 $ and $ \angle 4 $ are vertical angles → $ \angle 3 \cong \angle 4 $

But then why mention $ \angle 1 $ and $ \angle 2 $ being supplementary?

Perhaps $ \angle 2 $ is adjacent to $ \angle 1 $, and $ \angle 3 $ is opposite $ \angle 1 $, $ \angle 4 $ is opposite $ \angle 2 $

So:
- $ \angle 1 $ and $ \angle 4 $ are not adjacent — they are opposite?

No — “form a linear pair” means they are adjacent and on a straight line.

So $ \angle 1 $ and $ \angle 4 $ must be adjacent.

So likely:
- $ \angle 1 $ and $ \angle 4 $ are adjacent, on a straight line → $ \angle 1 + \angle 4 = 180^\circ $
- $ \angle 1 $ and $ \angle 2 $ are supplementary → $ \angle 1 + \angle 2 = 180^\circ $
- So $ \angle 2 = \angle 4 $
- $ \angle 2 $ and $ \angle 3 $ are vertical angles → $ \angle 2 \cong \angle 3 $
- So $ \angle 3 \cong \angle 4 $

Therefore, the proof should be:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 \cong \angle 4 $ | 4. Congruent Supplements Theorem |
| 5. $ \angle 3 \cong \angle 2 $ | 5. Vertical Angles Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property |

Yes — this makes sense.

So the completed table is:

| Statements | Reasons |
|-----------|--------|
| 1. $ \angle 1 $ and $ \angle 4 $ form a linear pair | 1. Given |
| 2. $ \angle 1 $ and $ \angle 4 $ are supplementary | 2. Supplement Theorem |
| 3. $ \angle 1 $ and $ \angle 2 $ are supplementary | 3. Given |
| 4. $ \angle 2 \cong \angle 4 $ | 4. Congruent Supplements Theorem |
| 5. $ \angle 3 \cong \angle 2 $ | 5. Vertical Angles Theorem |
| 6. $ \angle 3 \cong \angle 4 $ | 6. Transitive Property |

✔ Explanation: Since $ \angle 1 $ and $ \angle 4 $ are a linear pair, they are supplementary. $ \angle 1 $ and $ \angle 2 $ are also supplementary, so $ \angle 2 $ and $ \angle 4 $ are congruent (same supplement). $ \angle 3 $ and $ \angle 2 $ are vertical angles, so they are congruent. By transitivity, $ \angle 3 \cong \angle 4 $.

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✔ Final Answers:

#### Problem 11:
1. Given
2. Definition of Angle Bisector
3. Angle Addition Postulate
4. Substitution
5. Simplify
6. Division Property of Equality

#### Problem 12:
1. Given
2. Definition of Perpendicular Lines
3. Definition of Right Angle
4. Angle Addition Postulate
5. Substitution
6. Given
7. Definition of Congruent Angles
8. Substitution
9. Definition of Complementary Angles

#### Problem 13:
1. Given
2. Supplement Theorem
3. Given
4. $ \angle 2 \cong \angle 4 $
5. Vertical Angles Theorem
6. Transitive Property

> Note: In Problem 13, Step 4 should be: $ \angle 2 \cong \angle 4 $ — Congruent Supplements Theorem
> Step 5: $ \angle 3 \cong \angle 2 $ — Vertical Angles Theorem
> Step 6: $ \angle 3 \cong \angle 4 $ — Transitive Property

This completes all three proofs.