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 2

Angle relationships determine properties about triangles.

1 week
See Standards
Common Core: Major Standards:
G-CO.C.10

Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Clarification: Theorems include but are not limited to the listed theorems. Example: an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle

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

  • explore and prove qualities about the interior and exterior angles of any triangle
    • use examples and nonexamples to determine when an angle within a triangle can be classified as interior or exterior
    • analyze examples to justify that interior angles always add up to 180°  and the exterior angles add up to 360°  and attempt to create counterexamples
      • use tools to draw triangles and measure angles, and explain the impact of human error in this process
      • describe and classify triangles based on their angle measures (right, acute, obtuse)
    • describe how the relationships between interior angles of a triangle impact its corresponding side lengths and classification (equilateral, isosceles, scalene)
    • use deductive reasoning to prove that, for any triangle, the three interior angles add up to 180°  and the three exterior angles add up to 360°
      • draw or construct auxiliary lines and apply the properties of parallel lines
    • find the values of missing interior angles of a triangle
  • identify and describe relationships between the interior and exterior angles of a triangle
    • analyze examples to justify thatthe exterior angle is always equal to the sum of the two remote angles  and attempt to create counterexamples
    • use deductive reasoning to prove that the exterior angle is always equal to the sum of the two remote angles
      • describe the reflexive and transitive properties when using triangle sum and linear pair in the proof
    • find the measure of missing interior or exterior angles

Develop conceptual understanding:

Interior, exterior, triangle, isosceles, equilateral, scalene, right, acute, obtuse, example, nonexample, interior angles of a triangle, exterior angles of a triangle, remote interior angles (non-adjacent interior angles), auxiliary line, counterexample

Supporting terms to communicate:

side lengths, angle measure, congruent, construct, supplementary, complementary, congruent, straight angles, reflexive property, transitive property, substitute, adjacent
Core Resource

A core resource supports multiple days of instruction.

  • Analyzing Angles
    This incomplete core resource supports students in developing relationships between the measure of the interior, exterior, and remote angles of a triangle.
    Resource:
    Analyzing Angles

    Geometry & Algebra II Archive

    Geo U2: Proofs about Congruence

    Big Idea: Big Idea 2

    This incomplete core resource supports students in developing relationships between the measure of the interior, exterior, and remote angles of a triangle.

    Preview ResourceAdd a Copy of Resource to my Google Drive
    Type
    Geometry: Proofs About Congruence
    File
    Google Doc
    Tags
    angles, interior angles, triangles, exterior angles
    Standards
    Common Core: Major Standards ( G-CO.C.10 ) Common Core: Supporting Standards ( …
    +— Details
    Common Core: Major Standards:
    G-CO.C.10

    Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Clarification: Theorems include but are not limited to the listed theorems. Example: an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle

    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: Sharing Skepticism

The primary goal of this instructional routine is to support students in constructing and critiquing mathematical arguments.

  • Find the Value of an Angle
    Students will use what they know about angles in diagrams to find the value of a missing angle.
    Resource:
    Find the Value of an Angle

    Geometry & Algebra II Archive

    Geo U2: Proofs about Congruence

    Big Idea: Big Idea 2

    Students will use what they know about angles in diagrams to find the value of a missing angle.

    Preview ResourceAdd a Copy of Resource to my Google Drive
    Type
    Geometry: Proofs About Congruence
    File
    Google Doc
    Tags
    alternate interior angles, angles, construct viable argument, interior angles, instructional routine
    Standards
    Common Core: Major Standards ( G-CO.C.10 ) Common Core: Supporting Standards ( …
    +— Details
    Common Core: Major Standards:
    G-CO.C.10

    Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Clarification: Theorems include but are not limited to the listed theorems. Example: an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle

    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: