SOLUTION: Complete the formal proof of the theorem. If two line segments are congruent, then their midpoints separate these segments into four congruent segments. Line segment A B is abo

Question 1191305: Complete the formal proof of the theorem.
If two line segments are congruent, then their midpoints separate these segments into four congruent segments.
Line segment A B is above line segment D C. The two segments appear to be the same length. Points M and N are the midpoints of segments A B and D C, respectively.
Given:
AB ≅ DC
M is the midpoint of AB.
N is the midpoint of DC.
Prove:
AM ≅ MB ≅ DN ≅ NC
Statements Reasons
1.
AB ≅ DC
1.
Given

Correct: Your answer is correct.
2.
AB = DC

Correct: Your answer is correct.
2. If segments are ≅, then they are = in length.
3. AB = AM + MB;
AM + MB = DN + NC

Incorrect: Your answer is incorrect.
3.
Given

Incorrect: Your answer is incorrect.
4.
AM = MB = DN = NC

Incorrect: Your answer is incorrect.
4. Substitution
5.
M is the midpoint of AB;

N is the midpoint of DC.
5.
Given

Correct: Your answer is correct.
6.
AM = MB and DN = NC

Correct: Your answer is correct.
6. The midpoint of a segment forms two segments = in measure.
7. AM + AM = DN + DN 7.
Substitution

Correct: Your answer is correct.
8.
2 · AM = 2 · DN
8. Combine like terms.
9. AM = DN 9.
Division Property of Equality

Correct: Your answer is correct.
10.
AM + MB = DN + NC

Incorrect: Your answer is incorrect.
10. Substitution
11.
AM ≅ MB ≅ DN ≅ NC
11.
If segments are = in length, then they are ≅.

Correct: Your answer is correct.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's the completed proof with the corrected steps:
**Given:**
AB ≅ DC
M is the midpoint of AB.
N is the midpoint of DC.
**Prove:**
AM ≅ MB ≅ DN ≅ NC
| Statements | Reasons |
|---|---|
| 1. AB ≅ DC | 1. Given |
| 2. AB = DC | 2. If segments are ≅, then they are = in length. |
| 3. AM + MB = AB; DN + NC = DC | 3. Segment Addition Postulate |
| 4. AM + MB = DN + NC | 4. Substitution (since AB=DC) |
| 5. M is the midpoint of AB; N is the midpoint of DC. | 5. Given |
| 6. AM = MB and DN = NC | 6. The midpoint of a segment forms two segments = in measure. |
| 7. AM + AM = DN + DN | 7. Substitution (substituting MB for AM and NC for DN in step 4) |
| 8. 2 · AM = 2 · DN | 8. Combine like terms. |
| 9. AM = DN | 9. Division Property of Equality |
| 10. AM = MB and DN = NC (from step 6) and AM = DN (from step 9), therefore AM = MB = DN = NC | 10. Transitive Property of Equality |
| 11. AM ≅ MB ≅ DN ≅ NC | 11. If segments are = in length, then they are ≅. |

Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.

This request is about a SELF-EVIDENT statement.

Only out of boredom can one engage in such proofs.

RELATED QUESTIONS
Prove that if two intersecting line segments meet at their midpoints, the line segments... (answered by CPhill,Edwin McCravy)
Fundamental Ideas Points, Lines, and Planes Postulates and Theorems Segments,... (answered by richard1234)
I need help proving this theorem using a formal proof. If two inscribed angles... (answered by stanbon)
What is the converse of the following conditional? If two segments are congruent, then... (answered by avani)
I have two problems that I need help: Problem #1: Prove triangle ABC is congruent... (answered by venugopalramana)
What is the formal proof of the theorem "The diagonals of an isosceles trapezoid are... (answered by ikleyn)
10th Grade Geometry Proof Greetings... Here is the problem: Given: Line... (answered by jim_thompson5910)
The following figure shows a type of bridge truss called a Baltimore truss. If members... (answered by Theo)
EF is the median of trapezoid ABCD in the figure below. Use the following theorems to... (answered by greenestamps)