Proofs about Congruence

12 Components
Initial Task
What's My Angle?
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Analyzing Congruency Proofs
Re-engagement
Re-engagement for Unit 2
End of Unit Assessments
End of Unit Assessment for Unit 2 Performance Task for Unit 2 End of Unit Assessment Teacher Resources for GEO Unit 2 Find aligned Regents questions with Quiz Banker
Geo U2: Proofs about Congruence
Browse Components
Initial Task
What's My Angle?
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Analyzing Congruency Proofs
Re-engagement
Re-engagement for Unit 2
End of Unit Assessments
End of Unit Assessment for Unit 2 Performance Task for Unit 2 End of Unit Assessment Teacher Resources for GEO Unit 2 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 1

Relationships between angles determine whether lines are parallel.

1 week
See Standards
Common Core: Major Standards:
G-CO.C.9

Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Common Core: Supporting Standards:
G-CO.D.12

Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

Evidence of Understanding

  • describe the relationship between angles formed by two intersecting lines
    • use tools to explore and describe linear pairs and vertical angles
      • Discuss the impacts of human error
    • justify, using transformations and constructions, that vertical angles are congruent
    • use the angle addition postulate to show why supplementary angles sum to 180
    • deductively prove vertical angles must be congruent using the reflexive and transitive properties
      • explain and apply the subtraction property of equality (Euclid’s third common notion)
    • find the measure of an angle using complementary, supplementary, or vertical pairs
  • explore angle relationships formed by a transversal and a pair of parallel lines
    • use tools to make conjectures about parallel lines with a transversal
      • use constructions and transformations (and other tools) to map angles to each other
      • construct two congruent angles using a rigid motion transformation and parallel lines
      • measuring many pairs of angles and generalizing findings to justify when two angles are congruent or supplementary
    • prove that two lines are parallel using congruent angle relationships (the converse statement)
    • apply properties of vertical angles and linear pairs to prove that two angles are congruent or supplementary
    • create shared definitions for corresponding, alternate interior, alternate exterior, same-side interior or same-side exterior angles (also referred to as consecutive interior or consecutive exterior angles)
    • find the measure of a missing angle

Develop conceptual understanding:

linear pair, supplementary, vertical angles, reflexive property, transitive property, complementary, parallel, transversal, corresponding, alternate interior, alternate exterior, same-side (or consecutive) interior angles, same-side (or consecutive) exterior angles

Supporting terms to communicate:

adjacent, opposite, perpendicular, bisect, degree measure, human error, sum, equal measure, congruent, construct, rotate, translate, reflect, map, carry onto
Core Resource
A core resource supports multiple days of instruction.
  • Relationships Between Angles
    This multiple-day resource supports students in seeing relationships between lines and angles.
    Resource:
    Relationships Between Angles

    Geometry & Algebra II Archive

    Geo U2: Proofs about Congruence

    Big Idea: Big Idea 1

    The goal of these activities are to help students see that the ways lines and line segments are arranged with respect to each other lead to relationships between the angles formed by those lines and line segments.

    Preview ResourceAdd a Copy of Resource to my Google Drive
    Type
    Geometry: Proofs About Congruence
    File
    Google Doc
    Tags
    activity
    Standards
    Common Core: Major Standards ( G-CO.C.9 ) Common Core: Supporting Standards ( G…
    +— Details
    Common Core: Major Standards:
    G-CO.C.9

    Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

    Common Core: Supporting Standards:
    G-CO.D.12

    Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

    All Resources From:
Instructional Routine: Connecting Representations
These tasks are embedded within the instructional routine called Connecting Representations.
  • Angle Diagrams and Angle Theorems
    Students will be able to connect angle diagrams to proofs of angle theorems that use those diagrams.
    Resource:
    Angle Diagrams and Angle Theorems

    Geometry & Algebra II Archive

    Geo U2: Proofs about Congruence

    Big Idea: Big Idea 1

    Students will be able to connect angle diagrams to proofs of angle theorems that use those diagrams. 

    Preview ResourceAdd a Copy of Resource to my Google DriveLooking for Help with Instructional Routines? Learn More Here.
    Type
    Geometry: Proofs About Congruence
    File
    Google Doc
    Tags
    instructional routine, connecting representations
    Standards
    Common Core: Major Standards ( G-CO.C.9 ) Common Core: Supporting Standards ( G…
    +— Details
    Common Core: Major Standards:
    G-CO.C.9

    Prove geometric theorems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

    Common Core: Supporting Standards:
    G-CO.D.12

    Make geometric constructions. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Clarification: Constructions include but are not limited to the listed constructions. Example: constructing the median of a triangle or constructing an isosceles triangle with given lengths.

    All Resources From: