Proof Blueprint Anchor Chart
Geometry Blueprint Reference Guide
Two-Column Proof Anchor Chart
Logic & Reasoning
Algebraic to Geometric
1. Diagram Markings
Visual clues and symbols: tick marks (\(\cong\)), angle arcs, right angle boxes (\(90^\circ\)), and shared sides.
★ Code your diagram first!
2. Statements (WHAT)
Mathematical claims, equations, and congruence facts. Moving logically from Given to Prove.
Column 1: "What is true"
3. Reasons (WHY)
Authoritative laws justifying each claim: Definitions, Postulates, Properties, & Theorems.
Column 2: "Why it is true"
Algebraic Properties of Equality
Addition / Subtr. If \(a = b\), then \(a \pm c = b \pm c\)
Mult. / Division If \(a = b\), then \(ac = bc\) / \(\frac{a}{c} = \frac{b}{c}\)
Reflexive Prop. \(a = a\) or \(\overline{AB} \cong \overline{AB}\)
Symmetric Prop. If \(a = b\), then \(b = a\)
Transitive Prop. If \(a = b\) & \(b = c\), then \(a = c\)
Substitution If \(a = b\), replace \(a\) with \(b\)
Geometric Postulates & Definitions
Segment Addition \(AB + BC = AC\) (if \(B\) is between)
Angle Addition \(m\angle 1 + m\angle 2 = m\angle ABC\)
Def. of Midpoint Divides segment into \(2 \cong\) parts
Def. of \(\cong\) Segments \(AB = CD \iff \overline{AB} \cong \overline{CD}\)
Linear Pair Postulate Adjacent angles add to \(180^\circ\)
Vertical Angles Thm. Opposite angles are \(\cong\)
Model Proof: Overlapping Segment Theorem
Given: \(\overline{AB} \cong \overline{CD}\) | Prove: \(\overline{AC} \cong \overline{BD}\)
A B C D Given ≅ Shared Middle (Reflexive: BC = BC) Given ≅
| 1 Statements (What) | 2 Reasons (Why) |
|---|
| 1. \(\overline{AB} \cong \overline{CD}\) | 1. Given |
| 2. \(AB = CD\) | 2. Definition of Congruent Segments |
| 3. \(BC = BC\) | 3. Reflexive Property of Equality |
| 4. \(AB + BC = CD + BC\) | 4. Addition Property of Equality (add \(BC\) to both sides) |
| 5. \(AB + BC = AC\) and \(CD + BC = BD\) | 5. Segment Addition Postulate |
| 6. \(AC = BD\) | 6. Substitution Property of Equality |
| 7. \(\overline{AC} \cong \overline{BD}\) \(\blacksquare\) | 7. Definition of Congruent Segments |
4 Golden Rules for Mastering Proofs
Rule 1: Start & End
Step 1 is Given. The final step is always what you were asked to Prove.
Rule 2: = vs ≅
Use = for numerical values/lengths. Use ≅ for geometric shapes/figures.
Rule 3: Mark Diagrams
Always annotate your diagram with ticks, arcs, and shared segments before writing steps.
Rule 4: Valid Reasons
Reasons can NEVER be personal assumptions or "because it looks like it".
