Similarity Secrets Anchor ChartsSimilarity Secrets Geometric Relationships & Proportionality 01 Ratio & Proportion Ratio A comparison of two quantities by division. a : b or \(\frac{a}{b}\) Proportion An equation stating two ratios are equal. \(\frac{a}{b} = \frac{c}{d}\) Solving Tip: Cross-Multiplication If \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\) 02 Similar Polygons (\(\sim\)) "Same shape, different size." 1 Corresponding Angles are congruent (\(\cong\)). 2 Corresponding Sides are proportional (same ratio). ABCD WXYZ 03 Scale Factor (\(k\)) The ratio of any two corresponding lengths in similar figures. \(k = \frac{\text{New Length}}{\text{Original Length}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) Triangle Similarity The Shortcuts to Proving \(\triangle \sim \triangle\) 04 AA Similarity Angle-Angle If two angles of one triangle are congruent to two angles of another... \(\implies \text{Then the triangles are similar!}\) A 30° A 30° 05 SSS Similarity Side-Side-Side If the measures of all three pairs of corresponding sides are proportional... \(\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}\) 3, 4, 5 6, 8, 10 Ratio is constant (1:2) 06 SAS Similarity Side-Angle-Side If two pairs of corresponding sides are proportional AND their included angles are congruent... SAS \(\implies \triangle \sim \triangle\) 50° 4 6 50° 8 12 Mastering Similarity • Unit 04
Similarity Secrets Anchor Charts RevisedClassified Information Similarity Secrets Detective Guide to Proportional Figures FILE 01 Ratio & Proportion The Ratio A comparison of two quantities using division. \(a : b\) or \(\frac{a}{b}\) The Proportion An equation stating two ratios are equal. \(\frac{a}{b} = \frac{c}{d}\) The Cross-Product Rule To solve a proportion: \(ad = bc\) FILE 02 Similar Polygons (\(\sim\)) "Same shape, different size." A Corresponding Angles must be Congruent (\(\cong\)). S Corresponding Sides must be Proportional. P1 P2 Matching IDs Required FILE 03 Scale Factor (\(k\)) The constant ratio of corresponding side lengths. \(k = \frac{\text{Image}}{\text{Pre-image}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) Field Shortcuts Triangle Similarity The 3 "Shortcuts" to Prove Similarity FILE 04 AA Angle-Angle If two angles of one triangle are congruent to two angles of another, the triangles are similar. CRITERIA \(\angle A \cong \angle D\) and \(\angle B \cong \angle E\) ● ● FILE 05 SSS Side-Side-Side If the measures of all three pairs of corresponding sides are proportional, the triangles are similar. \(\frac{Side_1}{Side_1} = \frac{Side_2}{Side_2} = \frac{Side_3}{Side_3}\) 4 3 8 6 RATIO: 1:2 FILE 06 SAS Side-Angle-Side If two pairs of corresponding sides are proportional AND their included angles are congruent. The "Sandwich" Rule The angle must be between the sides! 10 8 20 16 GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-2026 STRICTLY EDUCATIONAL
Similarity Secrets Anchor Charts V2Classified Information Similarity Secrets Detective Guide to Proportional Figures Preliminary Case Brief: The Definition Similarity (\(\sim\)) Same shape, but a different size. Angles match, but sides are scaled. Congruence (\(\cong\)) Same shape AND the exact same size. An identical twin! RULE: All \(\cong\) figures are \(\sim\), but not all \(\sim\) figures are \(\cong\). FILE 01 Ratio & Proportion The Ratio \(a : b\) or \(\frac{a}{b}\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Cross-Product Rule To solve proportions: \(ad = bc\) FILE 02 Similar Polygons (\(\sim\)) A Angles must be Congruent (\(\cong\)). S Sides must be Proportional. #01 #01-B FILE 03 Scale Factor (\(k\)) \(k = \frac{\text{Image}}{\text{Pre-image}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) Field Shortcuts Triangle Similarity The 3 "Shortcuts" to Prove Similarity FILE 04 AA Angle-Angle If two angles of one triangle are congruent to two angles of another, the triangles are similar. CRITERIA \(\angle A \cong \angle D\) and \(\angle B \cong \angle E\) ● ● FILE 05 SSS Side-Side-Side If the measures of all three pairs of corresponding sides are proportional, the triangles are similar. \(\frac{Side_1}{Side_1} = \frac{Side_2}{Side_2} = \frac{Side_3}{Side_3}\) 4 3 8 6 RATIO: 1:2 FILE 06 SAS Side-Angle-Side If two pairs of corresponding sides are proportional AND their included angles are congruent. The "Sandwich" Rule The angle must be between the sides! 10 8 20 16 GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-2026 STRICTLY EDUCATIONAL
Similarity Secrets Anchor Charts V3Classified Information Similarity Secrets Phase I: Foundations of Proportionality Preliminary Case Brief Similarity (\(\sim\)) Same shape, but a different size. Angles match, sides are proportional. Congruence (\(\cong\)) Same shape AND the exact same size. Identical figures. FILE 01 Ratio & Proportion The Ratio \(a : b\) or \(\frac{a}{b}\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Cross-Product Rule: \(ad = bc\) To solve any proportion, multiply diagonally. FILE 02 Similar Polygons (\(\sim\)) A Corresponding Angles are congruent (\(\cong\)). S Corresponding Sides are proportional. FILE 03 Scale Factor (\(k\)) \(k = \frac{\text{New Length}}{\text{Old Length}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-A PAGE 1 OF 3 Field Shortcuts Triangle Similarity Phase II: Criteria for Proving Figures \(\sim\) FILE 04 AA Angle-Angle If two angles of one triangle are congruent to two angles of another, the triangles are similar. SHORTCUT \(\angle A \cong \angle D\), \(\angle B \cong \angle E\) ● ● FILE 05 SSS Side-Side-Side If all three pairs of corresponding sides have the same ratio, the triangles are similar. \(\frac{S_1}{S_1} = \frac{S_2}{S_2} = \frac{S_3}{S_3} = k\) 3 6 Proportional FILE 06 SAS Side-Angle-Side If two pairs of sides are proportional AND their included angle is congruent. The "Sandwich" The congruent angle must be between the proportional sides. GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-B PAGE 2 OF 3 Evidence Analysis Similarity in Action Phase III: Solving & Proportional Theorems FILE 07 Applying Properties How to solve for missing lengths: Identify Corresponding Parts. Set up a Proportion. Cross-multiply and solve. Example Setup
Similarity Secrets Anchor Charts V3 RevisedClassified Information Similarity Secrets Phase I: Foundations of Proportionality Preliminary Case Brief Similarity (\(\sim\)) Same shape, but a different size. Angles match, sides are proportional. Congruence (\(\cong\)) Same shape AND the exact same size. Identical figures. FILE 01 Ratio & Proportion The Ratio \(a : b\) or \(\frac{a}{b}\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Cross-Product Rule: \(ad = bc\) To solve any proportion, multiply diagonally across the equals sign. FILE 02 Similar Polygons (\(\sim\)) A Corresponding Angles are congruent (\(\cong\)). S Corresponding Sides are proportional (same ratio). ABCD WXYZ FILE 03 Scale Factor (\(k\)) \(k = \frac{\text{Image (New)}}{\text{Pre-image (Old)}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-A PAGE 1 OF 3 Field Shortcuts Triangle Similarity Phase II: Criteria for Proving Figures \(\sim\) FILE 04 AA Angle-Angle If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. SHORTCUT \(\angle A \cong \angle D\), \(\angle B \cong \angle E\) ● ● FILE 05 SSS Side-Side-Side If all three pairs of corresponding sides have the same ratio, the triangles are similar. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3} = k\) 4 8 Proportional FILE 06 SAS Side-Angle-Side If two pairs of sides are proportional AND their included angle is congruent. The "Sandwich" Rule The congruent angle must be between the proportional sides. GEOMETRY INVESTIGATION UNIT AGENT RECORD #SIM-04-B PAGE 2 OF 3 Evidence Analysis Similarity in Action Phase III: Solving & Proportional Theorems FILE 07 Applying Properties How to solve for missing lengths: Identify Corresponding Parts.
Similarity Secrets Anchor Charts V4 BWTop Secret / Classified Similarity Dossier Subject: Geometric Relationships & Proportionality Preliminary Case Brief Similarity (\(\sim\)) Same shape, but different size. Angles are congruent (\(\cong\)), but sides are proportional. Congruence (\(\cong\)) Identical in shape AND size. All corresponding parts are congruent. CASE FILE 01 Ratio & Proportion The Ratio \(a : b\) or \(\frac{a}{b}\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Condition: Cross-Product Property If \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\). Use this to solve for unknown variables. CASE FILE 02 Similar Polygons Required Conditions for Similarity: 1 Corresponding Angles are congruent (\(\cong\)). 2 Corresponding Sides are proportional. FIG A FIG B CASE FILE 03 Scale Factor (\(k\)) \(k = \frac{\text{Image Length}}{\text{Pre-Image Length}}\) Enlargement \(k > 1\) Reduction \(0 < k < 1\) REF #SIM-26-P1 SUBJECT: DOSSIER FOUNDATIONS Field Evidence Triangle Similarity Shortcuts to Prove Two Triangles are Similar THEOREMS 04 AA Similarity Angle-Angle Theorem: If two angles of one triangle are congruent (\(\cong\)) to two angles of another, then the triangles are similar. If \(\angle A \cong \angle D\) and \(\angle B \cong \angle E\), then \(\triangle ABC \sim \triangle DEF\) THEOREMS 05 SSS Similarity Side-Side-Side Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar. \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\) All Ratios Must Equal \(k\) THEOREMS 06 SAS Similarity Side-Angle-Side Theorem: If an angle of one triangle is \(\cong\) to an angle of another triangle, and the sides including those angles are proportional... \(\angle A \cong \angle D\) AND \(\frac{AB}{DE} = \frac{AC}{DF}\) SOLUTIONS 07 Applying Properties Case: Solving for Missing Evidence 1. List known segments. 2. Set up matching ratios. 3. Cross-multiply to solve.
Similarity Secrets Anchor Charts V4 BW RevisedConfidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles are congruent, sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact match in every dimension. Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Property: \(ad = bc\) (Cross-Product Rule) Chart 02 Conditions for Similarity 1. Corresponding Angles are congruent (\(\cong\)). 2. Corresponding Sides are proportional. Chart 03 Scale Factor (\(k\)) Calculation \(k = \frac{\text{New}}{\text{Old}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) File Ref: 404-SIM-01 Investigation Phase I: Polygons Field Evidence Triangle Similarity Shortcuts & Problem Solving Chart 04 AA Postulate Angle-Angle Condition: If 2 angles of one \(\triangle\) are \(\cong\) to 2 angles of another \(\triangle\), then the \(\triangle\)s are similar. Chart 05 SSS Theorem Side-Side-Side Condition: If all three corresponding side lengths are proportional, then the \(\triangle\)s are similar. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) Chart 06 SAS Theorem Side-Angle-Side Condition: Two pairs of proportional sides AND a congruent included angle. A S:S Chart 07 Solving for Parts The Strategy Mark Corresponding Sides. Setup a Proportion. Cross-multiply and solve. Indirect Measure 5 x \(\frac{5}{x} = \frac{\text{shadow}_1}{\text{shadow}_2}\) File Ref: 404-SIM-02 Investigation Phase II: Triangles Advanced Intel Proportionality Theorems Complex Geometric Relationships Chart 08 Triangle Proportionality Theorem: A line parallel to one side of a \(\triangle\) divides the other two sides proportionally.
Similarity Secrets Anchor Charts V4 BW Revised V2Confidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles match (\(\cong\)), sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact replica in every dimension. Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Cross-Product Condition: \(ad = bc\) Chart 02 Conditions for Similarity 1. Corresponding Angles are congruent (\(\cong\)). 2. Corresponding Sides are proportional. A B Chart 03 Scale Factor (\(k\)) Standard Formula \(k = \frac{\text{New}}{\text{Old}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) File Ref: 404-SIM-P1 Investigation Phase I: Polygons Field Evidence Triangle Similarity Methods for Proving Figures are Similar Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are congruent to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side The Condition: If all three pairs of corresponding sides are proportional, the triangles are similar. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) Chart 06 SAS Theorem Side-Angle-Side The Condition: Two pairs of proportional sides AND a congruent included angle. SIDE S A SIDE S Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement 5' Shadow A x Shadow B \(\frac{5}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation Phase II: Triangles Advanced Intel Proportionality Theorems Extended Laws of Similarity Chart 08
Similarity Secrets Anchor Charts V4 BW Revised V2 FinalConfidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles match (\(\cong\)), sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact replica in every dimension. Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Cross-Product Condition: \(ad = bc\) Chart 02 Conditions for Similarity 1. Corresponding Angles are congruent (\(\cong\)). 2. Corresponding Sides are proportional. A B Chart 03 Scale Factor (\(k\)) Standard Formula \(k = \frac{\text{New}}{\text{Old}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) File Ref: 404-SIM-P1 Investigation Phase I: Polygons Field Evidence Triangle Similarity Methods for Proving Figures are Similar Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are congruent to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 8 6 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement 5' Shadow A x Shadow B \(\frac{5}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation Phase II: Triangles Advanced Intel Proportionality Theorems Extended Laws of Similarity Chart 08 Triangle Proportionality
Similarity Secrets Anchor Charts V4 BW Revised V2 Final RevisedConfidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles match (\(\cong\)), sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact replica in every dimension. Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) A:B C:D Matched Ratios Cross-Product Condition: \(ad = bc\) Chart 02 Conditions for Similarity 1. Corresponding Angles are congruent (\(\cong\)). 2. Corresponding Sides are proportional. A B Chart 03 Scale Factor (\(k\)) Standard Formula \(k = \frac{\text{New}}{\text{Old}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) File Ref: 404-SIM-P1 Investigation Phase I: Polygons Field Evidence Triangle Similarity Methods for Proving Figures are Similar Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are congruent to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 8 6 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement 5' Shadow A x Shadow B \(\frac{5}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation Phase II: Triangles Advanced Intel Proportionality Theorems
Similarity Secrets Anchor Charts V4 BW Revised V2 Final Revised V2Confidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles match (\(\cong\)), sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact replica in every dimension. \(\cong\) \(\sim\) Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) A:B C:D Matched Ratios Cross-Product Condition: \(ad = bc\) Chart 02 Conditions for Similarity 1. Corresponding Angles are congruent (\(\cong\)). 2. Corresponding Sides are proportional. A B Chart 03 Scale Factor (\(k\)) Standard Formula \(k = \frac{\text{New}}{\text{Old}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) File Ref: 404-SIM-P1 Investigation Phase I: Polygons Field Evidence Triangle Similarity Methods for Proving Figures are Similar Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are congruent to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 8 6 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement 5' Shadow A x Shadow B \(\frac{5}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation Phase II: Triangles Advanced Intel
Similarity Secrets Anchor Charts V4 BW Revised V2 Final Revised V2 PolishConfidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, different Size. Angles match (\(\cong\)), sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. All corresponding parts match exactly. \(\cong\) \(\sim\) Chart 01 Ratio & Proportion Ratio \(a : b\) Proportion \(\frac{a}{b} = \frac{c}{d}\) Consistent Ratios Condition: \(ad = bc\) Chart 02 Similar Polygons Conditions for Figures to be Similar: 1. Corresponding ANGLES are \(\cong\). 2. Corresponding SIDES are Proportional. Chart 03 Scale Factor (\(k\)) Scale Factor Formula \(k = \frac{\text{New Length}}{\text{Old Length}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) 4 8 Ex: \(k = \frac{8}{4} = 2\) File Ref: 404-SIM-P1 Foundations: Polygons Field Evidence Triangle Similarity Criteria & Shortcut Theorems Chart 04 AA Postulate Angle-Angle Condition: If two angles of one triangle are \(\cong\) to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 12 9 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement 5' Shadow A x Shadow B \(\frac{5}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation: Triangles
Similarity Secrets Anchor Charts V4 BW Revised V2 Final Revised V2 Polish V2Confidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, but different Size. Angles match exactly, sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. All corresponding parts match perfectly. \(\cong\) \(\sim\) Chart 01 Ratio & Proportion Ratio \(a : b\) Proportion \(\frac{a}{b} = \frac{c}{d}\) Matched Proportions Condition: \(ad = bc\) Chart 02 Similar Polygons Conditions for Similarity: 1. Corresponding ANGLES are \(\cong\). 2. Corresponding SIDES are Proportional. Chart 03 Scale Factor (\(k\)) Scale Factor Formula \(k = \frac{\text{Image}}{\text{Pre-Image}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) 2 4 Example: \(k = \frac{4}{2} = 2\) File Ref: 404-SIM-P1 Foundations: Polygons Field Evidence Triangle Similarity Criteria & Shortcut Theorems Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are \(\cong\) to two angles of another, the triangles are similar (\(\sim\)). Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 12 9 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement Case 6 ft Shadow A x Shadow B \(\frac{6}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation: Triangle Proofs
Similarity Secrets Anchor Charts V4 BW Revised V2 Final Revised V2 Polish V3Confidential / Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Identical Shape, but different Size. Angles match exactly, sides are proportional. Congruence (\(\cong\)) Identical Shape AND Size. All corresponding parts match perfectly. \(\cong\) \(\sim\) Chart 01 Ratio & Proportion Ratio \(a : b\) Proportion \(\frac{a}{b} = \frac{c}{d}\) Matched Proportions Property: \(ad = bc\) Chart 02 Similar Polygons Conditions for Similarity: 1. Corresponding ANGLES are \(\cong\). 2. Corresponding SIDES are Proportional. Chart 03 Scale Factor (\(k\)) Scale Factor Formula \(k = \frac{\text{Image}}{\text{Pre-Image}}\) ENLARGEMENT \(k > 1\) REDUCTION \(k < 1\) 2 4 Example: \(k = \frac{4}{2} = 2\) File Ref: 404-SIM-P1 Foundations: Polygons Field Evidence Triangle Similarity Criteria & Shortcut Theorems Chart 04 AA Postulate Angle-Angle The Condition: If two angles of one triangle are \(\cong\) to two angles of another, the triangles are similar (\(\sim\)). Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding sides are proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 4 5 6 8 10 Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional sides and the included angle. 4 3 12 9 Chart 07 Solving for Missing Parts Strategy Blueprint Identify matching Corresponding Parts. Construct a Proportion equation. Cross-multiply and solve for the unknown. Indirect Measurement Case 6 ft Shadow A x Shadow B \(\frac{6}{x} = \frac{\text{Shadow A}}{\text{Shadow B}}\) File Ref: 404-SIM-P2 Investigation: Triangle Proofs
Similarity Secrets Anchor Charts V5 Final 4pagesConfidential Case Files Similarity Dossier Operation Proportionality: Sector 04 Preliminary Briefing Similarity (\(\sim\)) Identical Shape, but different Size. All corresponding angles are congruent, and all side lengths are proportional. Congruence (\(\cong\)) Identical Shape AND Size. An exact duplicate where every angle and side length is an identical match. Chart 01 Ratio & Proportion The Ratio \(a : b\) Comparison of 2 Values The Proportion \(\frac{a}{b} = \frac{c}{d}\) Equation of 2 Ratios The Cross-Product Rule: If \(\frac{a}{b} = \frac{c}{d}\) then \(ad = bc\) Chart 02 Similar Polygons 1 Corresponding Angles must be CONGRUENT (\(\cong\)). 2 Corresponding Sides must be PROPORTIONAL. A B REF: SIM-DOSSIER-P1 Investigation Phase I: Definitions Field Evidence Scale & Criteria Chart 03 Scale Factor (\(k\)) Master Formula \(k = \frac{\text{New}}{\text{Old}}\) Enlargement \(k > 1\) Reduction \(k < 1\) 3 9 \(k = \frac{9}{3} = 3\) Chart 04 AA Postulate Angle-Angle Condition: If TWO ANGLES of one triangle are \(\cong\) to two angles of another, the triangles are similar. Chart 05 SSS Theorem Side-Side-Side Condition: All three pairs of sides are proportional. \(\frac{A}{a} = \frac{B}{b} = \frac{C}{c}\) 4 4 2 8 8 4 REF: SIM-DOSSIER-P2 Investigation Phase II: Proofs Analysis Unit Solving Methods Chart 06 SAS Theorem Side-Angle-Side Condition: Two pairs of proportional sides AND the congruent included angle. 10 6 20 12 Chart 07 Solving for Missing Parts The Strategy Blueprint Match Corresponding Sides. Write the Proportion equation. Cross-multiply and solve for x. Application: Indirect Measurement 6' Shadow A
Similarity Dossier Practice PageActive Investigation Field Report: Similarity Case Agent: ____________________ Date: _________________________ Instructions: Analyze the evidence below. Show all calculations for verification by HQ. 01. Ratio Roundup Solve the following proportions for the unknown variable \(x\). A) \[ \frac{5}{12} = \frac{x}{36} \] x = ________ B) \[ \frac{x+2}{10} = \frac{4}{5} \] x = ________ 02. Similarity Suspects Identify the criterion (AA, SSS, or SAS) that proves the triangles are similar. If not similar, write "NONE". A) 40° 40° Criterion: ______________ B) 3 3 4 6 6 8 Criterion: ______________ 03. Field Operations Mission: A 6-foot tall detective casts a 4-foot shadow. At the same time, a nearby building casts a 20-foot shadow. Find the height of the building. Sketch your investigation evidence here... Calculations Height = __________ 04. Extended Intel Triangle Proportionality 6 3 10 x Find the value of \(x\) if the inner line is parallel to the base. x = ________ Scaling Law: Area Two similar rectangles have a scale factor of k = 4. If the area of the smaller rectangle is 15 cm², what is the area of the larger rectangle? Area = ________ File Ref: 404-SIM-PRAC Investigation Case Study: Final Assessment
Similarity Intelligence SummarySummary Intelligence Report Similarity Brief Essential Protocols & Theorems 01. Foundations Similarity (\(\sim\)): Same shape, proportional size. Angles \(\cong\), Sides proportional. Congruence (\(\cong\)): Exact match. Same shape AND size. Proportion: \(\frac{a}{b} = \frac{c}{d} \implies ad = bc\) 02. Scaling Law Scale Factor (\(k\)): \(k = \frac{\text{New Length}}{\text{Old Length}}\) Enlarge: \(k > 1\) Reduce: \(k < 1\) 03. Triangle Similarity Criteria AA 2 pairs of \(\cong\) angles. SSS 3 pairs of proportional sides. SAS 2 proportional sides + \(\cong\) included angle. Triangle Proportionality A parallel line divides the other two sides proportionally. \(\frac{A}{B} = \frac{C}{D}\) Transversal Proportions Parallel lines cut transversals into proportional segments. Ratio A = Ratio B 05. Dimensional Scaling Laws Linear (Perimeter) Ratio: \(k\) Area (Squared) Ratio: \(k^2\) REMINDER: Always verify that figures are confirmed similar before applying shortcuts. Similarity requires constant proportionality (\(k\)) across all corresponding elements. REF: SIM-SUMMARY-CHEAT Investigation Resource: Quick Reference
Similarity Dossier Answer KeyHQ Verification Only Answer Key: Field Report Master Reference Code: #SIM-KEY-2026 01. Ratio Roundup A) x = 15 Calculation: \(12x = 5 \cdot 36 \implies 12x = 180 \implies x = 15\) B) x = 6 Calculation: \(5(x+2) = 40 \implies 5x + 10 = 40 \implies 5x = 30 \implies x = 6\) 02. Similarity Suspects A) Criterion: AA Reasoning: Both triangles have a right angle and a 40° angle. B) Criterion: SSS Reasoning: Sides have constant ratio of 1:2 (\(3/6 = 3/6 = 4/8 = 0.5\)). 03. Field Operations Building Height: 30 feet Setup: \(\frac{\text{Height}}{\text{Shadow}} \implies \frac{6}{4} = \frac{x}{20}\) Solve: \(4x = 120 \implies x = 30\) 04. Extended Intel A) x = 5 Setup: \(\frac{6}{3} = \frac{10}{x} \implies 6x = 30 \implies x = 5\) B) Area = 240 cm² Law: New Area = Old Area \(\cdot k^2\) Math: \(15 \cdot 4^2 = 15 \cdot 16 = 240\) File Ref: 404-SIM-KEY HQ VERIFICATION COMPLETE
Similarity Secrets Anchor Charts V6 Fixed 4pagesConfidential Case Files Similarity Dossier Geometric Investigation: Unit 04 Initial Briefing Similarity (\(\sim\)) Same shape, different size. Sides are proportional. Congruence (\(\cong\)) Identical copy. Same shape AND same size. \(\cong\) \(\sim\) Chart 01 Ratio & Proportion The Ratio \(a : b\) The Proportion \(\frac{a}{b} = \frac{c}{d}\) Condition: \(ad = bc\) The Cross-Product Rule Chart 02 Similar Polygons 1 Angles are Congruent (\(\cong\)). 2 Sides are Proportional. FIG A FIG B REF: SIM-DOS-01 Investigation Phase I Analysis Hub Criteria & Scale Chart 03 Scale Factor (\(k\)) Calculation Protocol \(k = \frac{\text{New Length}}{\text{Old Length}}\) ENLARGE: \(k > 1\) REDUCE: \(k < 1\) 4 12 Scale = 3 Chart 04 AA Postulate Angle-Angle Condition: If TWO ANGLES of one triangle match two angles of another, the triangles are similar (\(\sim\)). ● ● Chart 05 SSS Theorem Side-Side-Side Condition: All corresponding side pairs proportional. \(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3}\) 3 3 4 6 6 8 REF: SIM-DOS-02 Investigation Phase II Evidence Unit Solving Unit Chart 06 SAS Theorem Side-Angle-Side Condition: 2 proportional side pairs and the congruent INCLUDED ANGLE between them. 10 6 20 12 Chart 07 Solving Protocol The Blueprint Identify Matching Sides. Construct Proportion. Cross-Multiply & Solve. 6 FT X \(\frac{6}{X} = \frac{\text{SHADOW A}}{\text{SHADOW B}}\) Chart 08 Triangle Proportionality Theorem: A line parallel to a side divides the other two sides proportionally.
