Question: User: Advanced Geometry
Practice 2-7 Proving Segment Relationships
Name: Pepper Heaton
OBJECTIVE: Students will write proofs involving segment addition and segment congruence.
Proof #1
Given: $LK \cong NM$, $KJ \cong MJ$
Prove: $LJ \cong NJ$
Statements
1. $LK \cong NM$ and $KJ \cong MJ$
2. $LK = NM$ and $KJ = MJ$
3. $LK + KJ = NM + MJ$
4. $LJ = NM + MJ$
5. $LJ = NJ$
6. $LJ \cong NJ$
Reasons
1. Given
2. Definition of congruent segments
3. Substitution Property
4. Segment Addition Postulate
5. Transitive Property
6. Definition of congruent segments
Proof #2
Given: $C$ is the midpoint of $AE$.
$C$ is the midpoint of $BD$.
$AE \cong BD$
Prove: $AC \cong CD$
Proof:
Statements
1. $C$ is the midpoint of $AE$, $BD$, $AE \cong BD$
2. $AC = CE$, $BC = CD$
3. $AE = BD$
4. $AE = AC + CE$, $BD = BC + CD$
5. $AC + CE = BC + CD$
6. $AC + AC = CD + CD$
7. $2AC = 2CD$
8. $AC = CD$
9. $AC \cong CD$
Reasons
1. Given
2. Definition of midpoint
3. Definition of congruent segments
4. Segment Addition Postulate
5. Substitution Property
6. Substitution Property
7. Simplify
8. Division Property
9. Definition of congruent segments
Graph shapes and math:
top-right: a triangle-like figure with points $J$, $K$, $L$, $M$, $N$, showing segments $JK$, $KL$, $JM$, $MN$, and $NL$; the given congruent segments are $LK \cong NM$ and $KJ \cong MJ$, leading to $LJ \cong NJ$.
center: an X-shaped intersecting-segments figure with $A$ at top-left, $B$ at top-right, $D$ at bottom-left, $E$ at bottom-right, and $C$ at the intersection; $C$ is the midpoint of $AE$ and $BD$, and the goal is $AC \cong CD$.
1. **State the problem:**
We have two proofs involving segment congruence and segment addition.
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### Proof #1
**Given:** $LK \cong NM$, $KJ \cong MJ$
**Prove:** $LJ \cong NJ$
2. **Recall definitions and properties:**
- Congruent segments have equal lengths: $AB \cong CD \Rightarrow AB = CD$.
- Segment Addition Postulate: If $B$ is between $A$ and $C$, then $AB + BC = AC$.
- Transitive Property: If $a = b$ and $b = c$, then $a = c$.
3. **Step-by-step proof:**
- From the given, $LK \cong NM$ and $KJ \cong MJ$.
- By definition of congruent segments, $LK = NM$ and $KJ = MJ$.
- Add these equal segments: $LK + KJ = NM + MJ$.
- By the Segment Addition Postulate, $LJ = LK + KJ$ and $NJ = NM + MJ$.
- Substitute to get $LJ = NJ$.
- By definition of congruent segments, $LJ \cong NJ$.
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### Proof #2
**Given:** $C$ is the midpoint of $AE$ and $BD$, and $AE \cong BD$.
**Prove:** $AC \cong CD$
4. **Recall definitions and properties:**
- Midpoint divides a segment into two equal parts: $C$ midpoint of $AE$ means $AC = CE$.
- Congruent segments have equal lengths.
- Segment Addition Postulate.
- Simplification and division properties.
5. **Step-by-step proof:**
- Given $C$ is midpoint of $AE$ and $BD$, so $AC = CE$ and $BC = CD$.
- Given $AE \cong BD$, so $AE = BD$.
- By Segment Addition Postulate, $AE = AC + CE$ and $BD = BC + CD$.
- Substitute equal parts: $AC + CE = BC + CD$.
- Since $AC = CE$ and $BC = CD$, rewrite as $AC + AC = CD + CD$.
- Simplify: $2AC = 2CD$.
- Divide both sides by 2: $AC = CD$.
- By definition of congruent segments, $AC \cong CD$.
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**Final answers:**
- Proof #1: $LJ \cong NJ$
- Proof #2: $AC \cong CD$