Proving Segment Relationships
Put your proof skills to work on segments — using the Segment Addition Postulate, midpoint definition, and properties of equality to prove congruence.
Introduction
Now that you know the proof format, it's time to prove statements about segment lengths. The key tools are Segment Addition, definitions of midpoint and congruence, and the properties of equality.
Past Knowledge
Two-column proofs (2.3.3). Segment Addition (1.2.1). Midpoint (1.2.2).
Today's Goal
Write two-column proofs involving segment congruence and length relationships.
Future Success
Triangle congruence proofs (Unit 5) often begin with segment relationships.
Key Concepts
Key Theorems for Segment Proofs
Segment Congruence
if and only if .
Proof Toolkit
- Segment Addition Postulate: If is between and , then
- Definition of Midpoint: is the midpoint →
- Definition of Congruent Segments: Equal lengths ↔ congruent segments
- Substitution, Transitive, Symmetric Properties
Worked Examples
Overlapping Segments
Given: . Prove: . (Points in order A, B, C, D on a line.)
| Statements | Reasons |
|---|---|
| Given | |
| Segment Addition Post. | |
| Segment Addition Post. | |
| Substitution | |
| Subtraction Property of Equality |
Midpoint Congruence
Given: is the midpoint of and is the midpoint of . . Prove: .
| Statements | Reasons |
|---|---|
| is midpoint of ; is midpoint of | Given |
| ; | Def. of midpoint |
| Given | |
| Division Property of Equality | |
| Substitution |
Algebraic Segment Proof
Given: , , . B is between A and C. Find and prove .
| Statements | Reasons |
|---|---|
| Segment Addition Post. | |
| Substitution | |
| Simplify | |
| Subtr. & Div. Properties | |
| , | Substitution |
Result: , so . B is between A and C but is not the midpoint — a common trick question!
Common Pitfalls
Mixing Up Congruence and Equality
Use for measures (numbers). Use for segments themselves. Don't write ; write or .
Real-Life Applications
Construction & Engineering
When building symmetric structures like bridges, engineers must prove that opposing beams are equal in length. They use the same logical reasoning — segment addition and congruence — to verify structural integrity.
Practice Quiz
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