Solved Name: Unit 2: Logic & Proof Homework 8: Angle Proofs ... - Free Printable

Let's solve each of these geometry proofs step by step, filling in the missing statements and reasons. We'll go through problems 11, 12, and 13 one at a time.

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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. Combine like terms |
| 6. $m\angle MKL = \frac{1}{2} m\angle JKL$ | 6. Division Property of Equality |

✔ Explanation: Since $\overline{KM}$ bisects $\angle JKL$, it splits the angle into two equal parts. So, $\angle JKM$ and $\angle MKL$ are congruent. Using the angle addition postulate and substitution, we show that twice the measure of $\angle MKL$ equals the whole angle, so $\angle MKL$ must be half of $\angle JKL$.

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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 $90^\circ$) |

✔ Explanation: Since $\overline{BD} \perp \overline{BC}$, $\angle DBC = 90^\circ$. The angles $\angle DBE$ and $\angle EBC$ add up to this right angle. But $\angle ABD$ is congruent to $\angle DBE$, so replacing $\angle DBE$ with $\angle ABD$ gives us that $\angle ABD + \angle EBC = 90^\circ$, meaning 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. $m\angle 1 + m\angle 4 = 180^\circ$ | 2. Supplement Theorem (linear pairs are supplementary) |
| 3. $m\angle 1 + m\angle 2 = 180^\circ$ | 3. Given ($\angle 1$ and $\angle 2$ are supplementary) |
| 4. $m\angle 4 = m\angle 2$ | 4. Congruent Supplements Theorem (if two angles are supplementary to the same angle, they are congruent) |
| 5. $\angle 2 \cong \angle 3$ | 5. Vertical Angles Theorem (assuming $\angle 2$ and $\angle 3$ are vertical angles — implied by diagram) |
| 6. $\angle 3 \cong \angle 4$ | 6. Transitive Property of Congruence |

✔ Explanation:
- From the given, $\angle 1$ and $\angle 4$ form a linear pair → they are supplementary.
- Also, $\angle 1$ and $\angle 2$ are supplementary.
- So, both $\angle 4$ and $\angle 2$ are supplementary to $\angle 1$ → by the Congruent Supplements Theorem, $\angle 4 \cong \angle 2$.
- Then, if $\angle 2$ and $\angle 3$ are vertical angles (as suggested by the typical diagram), they are congruent.
- Therefore, by transitivity, $\angle 3 \cong \angle 4$.

> ⚠️ Note: In step 5, the reason "Vertical Angles Theorem" assumes that $\angle 2$ and $\angle 3$ are vertical angles. This is standard in such diagrams where two lines intersect.

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✔ Final Answers:

All three proofs are now complete with correct statements and reasons. Each uses fundamental geometric definitions and theorems:
- Angle Bisector
- Perpendicular Lines
- Linear Pairs & Supplementary Angles
- Vertical Angles
- Transitive Property

Let me know if you'd like a visual explanation or help drawing the diagrams!