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Pre-AP Geometry Assessment Module 1- Proofs on Congruence and Parallelograms Name:______________________________________________ Date:_______________ Period:_____ Directions: For each proof, mark the diagram with both the given evidence along with any additional evidence needed to support the proof. Correctly order the proof by placing numbers in the table beside each line. The evidence and the justification are correct for each line! 1. Given: Prove: ̅̅̅̅ ⊥ 𝑪𝑫 ̅̅̅̅, 𝑨𝑩 ̅̅̅̅ ⊥ 𝑨𝑫 ̅̅̅̅ , 𝒎∠𝟏 = 𝒎∠𝟐 𝑩𝑪 △ 𝑩𝑪𝑫 ≅ △ 𝑩𝑨𝑫 Rigid transformation that most likely maps the two triangles: _____________________________________________________________ Number for Proof Order Evidence Justification 𝒎∠𝑪𝑫𝑩 = 𝒎∠𝑨𝑫𝑩 If two angles are equal in measure, then their supplements are equal in measure. 𝒎∠𝟏 = 𝒎∠𝟐 Given 𝒎∠𝟏 + 𝒎∠𝑪𝑫𝑩 = 𝟏𝟖𝟎° Linear pairs form supplementary angles. △ 𝑩𝑪𝑫 ≅ △ 𝑩𝑨𝑫 AAS 𝒎∠𝑩𝑪𝑫 = 𝒎∠𝑩𝑨𝑫 = 𝟗𝟎° Definition of perpendicular line segments 𝒎∠𝟐 + 𝒎∠𝑨𝑫𝑩 = 𝟏𝟖𝟎° Linear pairs form supplementary angles. ̅̅̅̅ 𝑨𝑩 ⊥ ̅̅̅̅ 𝑨𝑫 Given 𝑩𝑫 = 𝑩𝑫 Reflexive property ̅̅̅̅ ⊥ 𝑪𝑫 ̅̅̅̅ 𝑩𝑪 Given Diagram Given QED ̅̅̅̅ and 𝑹𝒀 ̅̅̅̅ are the 2. Given: In the figure, 𝑹𝑿 ̅̅̅̅ perpendicular bisectors of 𝑨𝑩 and ̅̅̅̅ 𝑨𝑪, respectively. Prove: (a) △ 𝑹𝑨𝑿 ≅ △ 𝑹𝑩𝑿 (b) ̅̅̅̅ 𝑹𝑨 ≅ ̅̅̅̅ 𝑹𝑩 ≅ ̅̅̅̅ 𝑹𝑪 Rigid transformation that most likely maps the two triangles in part a: _____________________________________________________________ Number for Proof Order Evidence Justification ̅̅̅̅ ≅ 𝑿𝑩 ̅̅̅̅ 𝑨𝑿 Definition of segment bisector QED ̅̅̅̅ 𝑹𝑨 ≅ ̅̅̅̅ 𝑹𝑩 ≅ ̅̅̅̅ 𝑹𝑪 Transitive property △ 𝑹𝑨𝒀 ≅ △ 𝑹𝑪𝒀 SAS △ 𝑹𝑨𝑿 ≅ △ 𝑹𝑩𝑿 SAS ̅̅̅̅ 𝑹𝒀 is the perpendicular ̅̅̅̅ bisector of 𝑨𝑪 Given 𝒎∠𝑹𝒀𝑨 = 𝟗𝟎°, 𝒎∠𝑹𝒀𝑪 = 𝟗𝟎° Definition of perpendicular bisector ̅̅̅̅ ̅̅̅̅ ≅ 𝒀𝑪 𝑨𝒀 Definition of segment bisector ̅̅̅̅ 𝑹𝑿 is the perpendicular ̅̅̅̅. bisector of 𝑨𝑩 Given 𝒎∠𝑹𝑿𝑨 = 𝟗𝟎°, 𝒎∠𝑹𝑿𝑩 = 𝟗𝟎° Definition of perpendicular bisector ̅̅̅̅ ≅ 𝑹𝑿 ̅̅̅̅ 𝑹𝑿 Reflexive property ̅̅̅̅ 𝑹𝒀 ≅ ̅̅̅̅ 𝑹𝒀 Reflexive property ̅̅̅̅ ̅̅̅̅ ≅ 𝑹𝑩 ̅̅̅̅, 𝑹𝑨 ̅̅̅̅ ≅ 𝑹𝑪 𝑹𝑨 CPCTC Diagram Given 3. Given: Square 𝑨𝑩𝑪𝑺 ≅ Square 𝑬𝑭𝑮𝑺, ⃡𝑹𝑨𝑩, ⃡𝑹𝑬𝑭 Prove: △ 𝑨𝑺𝑹 ≅ △ 𝑬𝑺𝑹 Rigid transformation that most likely maps the two triangles: _____________________________________________________________ Number for Proof Order Evidence Justification ∠𝑩𝑨𝑺 and ∠𝑭𝑬𝑺 are right angles. Definition of square 𝑨𝑺 = 𝑬𝑺 CPCTC Square 𝑨𝑩𝑪𝑺 ≅ Square 𝑬𝑭𝑮𝑺 Given QED ∠𝑩𝑨𝑺 and ∠𝑺𝑨𝑹 form a linear pair. Definition of linear pair ∠𝑭𝑬𝑺 and ∠𝑺𝑬𝑹 form a linear pair. Definition of linear pair △ 𝑨𝑺𝑹 ≅ △ 𝑬𝑺𝑹 HL ∠𝑺𝑨𝑹 and ∠𝑺𝑬𝑹 are right angles. Two angles that are supplementary and congruent each measure 𝟗𝟎° and are, therefore, right angles. △ 𝑨𝑺𝑹 and △ 𝑬𝑺𝑹 are right triangles. △ 𝑨𝑺𝑹 and △ 𝑬𝑺𝑹 are right triangles. Diagram Given 𝑺𝑹 = 𝑹𝑺 Reflexive property 4 v e n 4. : Given: Parallelogram 𝑮𝑯𝑰𝑱 with diagonals of equal length, 𝑮𝑰 = 𝑯𝑱 G H J I Prove: GHIJ is a rectangle Number for Proof Order Evidence Justification Diagram Given QED 𝑮𝑯𝑰𝑱 is a rectangle. Definition of a rectangle 𝒎∠𝑮 = 𝒎∠𝑱 = 𝒎∠𝑯 = 𝒎∠𝑰 = 𝟗𝟎° Substitution property of equality △ 𝑯𝑱𝑮 ≅ △ 𝑰𝑮𝑱, SSS △ 𝑮𝑯𝑰 ≅ △ 𝑱𝑰𝑯 𝒎∠𝑮 + 𝒎∠𝑱 = 𝟏𝟖𝟎°, 𝒎∠𝑯 + 𝒎∠𝑰 = 𝟏𝟖𝟎° If parallel lines are cut by a transversal, then interior angles on the same side are supplementary. Parallelogram 𝑮𝑯𝑰𝑱 with diagonals of equal length, Given 𝑮𝑰 = 𝑯𝑱 𝑮𝑱 = 𝑱𝑮, 𝑯𝑰 = 𝑰𝑯 Reflexive property 𝑮𝑯 = 𝑰𝑱 Opposite sides of a parallelogram are congruent. 𝒎∠𝑮 = 𝒎∠𝑱, 𝒎∠𝑯 = 𝒎∠𝑰 CPCTC 𝒎∠𝑮 = 𝟗𝟎°, Division property of equality 𝒎∠𝑯 = 𝟗𝟎°
