Geometry Proofs Reference Guide
Washington State Learning Standards • Mathematics (CCSS-M)
Standard Deep Dive: WA.HSG-CO.B.8
Unpacked Standards Guide
Embedded DOK Learning Framework
Standard Text: Explain how the criteria for triangle congruence (\(\text{ASA}\), \(\text{SAS}\), \(\text{SSS}\)) follow from the mathematical definition of congruence in terms of rigid motions.
Prerequisite Skills & Foundational Standards
Grade 8 • Transformations Progression
8.G.A.2: Rigid Motions
Understand that 2D figures are congruent if one maps to the other via rotations, reflections, and translations.
HSG-CO.A.5: Mapping
Given a figure and a rigid motion, draw transformed images and specify sequences of transformations connecting shapes.
HSG-CO.B.7: Superposition
Use rigid motions to show figures are congruent if and only if all corresponding pairs of sides and angles coincide.
Deconstructed Learning Targets with Embedded DOK Levels
Cognitive Progression: Levels 1 → 3
LT 1: Terminology & Identification DOK 1: Recall
“I can accurately identify corresponding sides and angles between geometric figures, state the five congruence criteria (\(\text{SSS}\), \(\text{SAS}\), \(\text{ASA}\), \(\text{AAS}\), \(\text{HL}\)), and recall that rigid motions preserve segment lengths and angle measures.”
Cognitive Focus: Identify, State, Match Foundational
LT 2: Classification & Mapping DOK 2: Skills & Concepts
“I can determine which congruence criterion applies given marked geometric diagrams, identify any missing congruent components required, and describe a precise sequence of rigid motions mapping one triangle onto another.”
Cognitive Focus: Classify, Map, Apply Application
LT 3: Reasoning & Counterexamples DOK 3: Strategic Thinking
“I can justify mathematically why the SAS and ASA criteria guarantee triangle congruence based on rigid-motion superposition, and construct explicit counterexamples (such as the swinging side) to explain why SSA and AAA fail.”
Cognitive Focus: Justify, Critique, Analyze Justification
LT 4: Deductive Synthesis & CPCTC DOK 3: Complex Proof
“I can construct multi-step formal proofs using congruence criteria combined with definitions, reflexive/transitive properties, and parallel angle theorems, then apply CPCTC to prove relationships between geometric parts.”
Cognitive Focus: Synthesize, Prove, Deduce Formal Proof
Observable Success Criteria by DOK Level
- DOK 1: Correctly names corresponding vertices, sides, and angles in formal congruent statements (\(\Delta ABC \cong \Delta DEF\)).
- DOK 2: Accurately distinguishes between included angles (\(\text{SAS}\)) and non-included angles (\(\text{SSA}\)) in given diagrams.
- DOK 3: Demonstrates that rigid motions preserve distances and angles to prove overlapping parts coincide without empirical measurement.
- DOK 3: Sequentially chains proof steps so CPCTC is invoked strictly after establishing triangle congruence.
Diagnostic Misconceptions & OSPI Assessment
- SSA Fallacy (DOK 2/3): Assuming two sides and a non-included angle fix a triangle. Remedy: Sketch obtuse vs acute ambiguous triangles.
- Recipe vs. Rigor: Reciting letters (\(\text{SAS}\)) without recognizing isometry foundations. Remedy: Model physical patty-paper superposition.
- OSPI Smarter Balanced Claim 3: Washington state assessments heavily weight Claim 3 (Communicating Mathematical Reasoning) through written justifications.
Washington State Learning Standards • Mathematics (CCSS-M)
High School Geometry: Congruence & Proofs
Domain: HSG-CO
Grades 9–11 Core Reference
WA.HSG-CO.B.8
DOK 1–3
Explain how criteria (\(\text{ASA}\), \(\text{SAS}\), \(\text{SSS}\)) follow from rigid motions. (See Page 1 for full targets & criteria).
WA.HSG-CO.C.9
Prove theorems about lines and angles: vertical angles, parallel lines with transversals, and perpendicular bisector equidistance.
WA.HSG-CO.C.10
Prove theorems about triangles: interior angle sum \(180^\circ\), isosceles base angles, and the triangle midsegment theorem.
The Five Valid Triangle Congruence Criteria
Rigid Motions Preserving Distance & Angle
SSS Side-Side-Side
3 pairs of corresponding sides are congruent.
SAS Side-Angle-Side
2 sides and the included angle are congruent.
ASA Angle-Side-Angle
2 angles and the included side are congruent.
AAS Angle-Angle-Side
2 angles and an adjacent non-included side.
HL Hypotenuse-Leg
Hypotenuse and 1 leg congruent in right \(\Delta\) only.
Critical Caution: \(\text{SSA}\) (or \(\text{ASS}\)) and \(\text{AAA}\) are never valid criteria to prove congruence! \(\text{SSA}\) is ambiguous, and \(\text{AAA}\) proves similarity only.
Essential Proof Bank (Lines, Angles, and Triangles)
Vertical Angles Theorem HSG-CO.C.9
When two straight lines intersect, the opposite non-adjacent angles are congruent: \(\angle 1 \cong \angle 3\).
Parallel Line Postulate & Theorems HSG-CO.C.9
If \(l \parallel m\) cut by transversal \(t\): Alternate Interior \(\angle s \cong\), Corresponding \(\angle s \cong\), Consecutive Interior \(\angle s\) supplementary.
Triangle Angle Sum Theorem HSG-CO.C.10
The interior angles of any planar triangle sum to exactly \(180^\circ\). Proved by drawing an auxiliary line parallel to a base.
Isosceles Triangle Base Angles HSG-CO.C.10
If two sides of a triangle are congruent, the angles opposite them are congruent (\(\overline{AB} \cong \overline{AC} \implies \angle B \cong \angle C\)).
Triangle Midsegment Theorem HSG-CO.C.10
A segment connecting the midpoints of two sides is parallel to the third side and half its length (\(DE \parallel BC\) and \(DE = \frac{1}{2}BC\)).
Anatomy of a Geometric Proof & Valid Justifications
CPCTC Rule
1. Definitions Midpoint, angle bisector, perpendicular lines, complementary/supplementary angles.
2. Properties Reflexive (\(\overline{AB} \cong \overline{AB}\)), Symmetric, Transitive, and Substitution properties of equality.
3. Postulates/Criteria Segment addition, angle addition, parallel postulate, \(\text{SAS}\), \(\text{SSS}\), \(\text{ASA}\), \(\text{AAS}\), \(\text{HL}\).
4. CPCTC Corresponding Parts of Congruent Triangles are Congruent (used after proving \(\Delta \cong\)).
Washington State Learning Standards • Mathematics (CCSS-M)
High School Geometry: Similarity & Applications
Domains: HSG-SRT & HSG-MG
Modeling & Trigonometric Foundations
WA.HSG-SRT.A.2
Define similarity in terms of dilations and rigid transformations; verify angle preservation and side proportionality.
WA.HSG-SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and prove geometric relationships.
WA.HSG-MG.A.1, 3
Apply geometric methods to solve design problems, indirect measurement, and physical structures.
Triangle Similarity Criteria (\(\sim\))
Preserves Shape • Scale Factor \(k \neq 1\)
AA \(\sim\) Criterion
Angle-Angle Similarity
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
\(\angle A \cong \angle D \land \angle B \cong \angle E \implies \Delta ABC \sim \Delta DEF\)
SAS \(\sim\) Criterion
Side-Angle-Side Similarity
Two pairs of sides are proportional and the included angles between them are congruent.
\(\frac{AB}{DE} = \frac{AC}{DF} = k \land \angle A \cong \angle D\)
SSS \(\sim\) Criterion
Side-Side-Side Similarity
All three pairs of corresponding sides are in the same constant ratio (scale factor \(k\)).
\(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k\)
Key Similarity Theorems & Proportionality Ratios
Triangle Proportionality (Side-Splitter) HSG-SRT.B.4
If a line is parallel to one side of a triangle intersecting the other two, it divides them proportionally: \(\frac{AD}{DB} = \frac{AE}{EC}\).
Right Triangle Altitude (Geometric Mean) HSG-SRT.B.5
The altitude to the hypotenuse divides the triangle into two triangles similar to each other and the original: \(h^2 = x \cdot y\).
Dimensional Scaling Relationships (Length, Area, Volume) HSG-MG.A.1
Linear Ratio: \(k = \frac{\text{length}_2}{\text{length}_1}\)
Area Ratio: \(k^2 = \frac{\text{Area}_2}{\text{Area}_1}\)
Volume Ratio: \(k^3 = \frac{\text{Vol}_2}{\text{Vol}_1}\)
Applied Geometry in Washington State Curricula (HSG-MG)
Smarter Balanced Target P
Indirect Measurement
Using shadow reckoning and mirrors with \(AA \sim\) to calculate tree, tower, or flagpole heights inaccessible to tape measures.
Truss Rigidity & Architecture
Explaining why bridge and roof trusses use triangles: by \(\text{SSS}\) rigidity, fixed side lengths uniquely fix all interior angles.
Scale Models & Cartography
Utilizing scale factors on topographic maps and blueprints, converting scale distances and calculating land surface areas via \(k^2\).
WA State OSPI Assessment Tip: High school proof items require students to provide explicit mathematical justifications (theorem names or definitions) at each step.
