Unit 1

Congruence

Geometry

 

Suggested Unit Length and Description:

 

45 days

 

In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. (Mathematics Appendix A, p.29)

 

·         Experiment with transformations in the plane

·         Understand congruence in terms of rigid motions

·         Prove geometric theorems

·         Make geometric constructions

 

Standards:

 

G-CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

 

G-CO.A.2 Represent transformations in the plane using, e.g. transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

 

G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

 

G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

 

G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

 

G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

 

G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

 

G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

 

G-CO.C.9 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.

 

G-CO.C.10 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.

 

G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

·         Standards Clarification: The criteria for triangle congruence will be used to prove theorems about parallelograms. Students will continue to find the distance between points in order to calculate segment lengths or perimeter of figures as they focus more heavily on analytic proofs in this unit.

 

G-CO.D.12 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.

 

G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

 

G-CO.A.1

·         I can describe the undefined terms: point, line, and distance along a line in a plane.

·         I can define circle and the distance around a circular arc.

G-CO.A.2

·         I can describe the different types of transformations including translations, reflections, rotations and dilations.

·         I can describe transformations as functions that take points in the coordinate plane as inputs and give other points as outputs.

·         I can compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

·         I can represent transformations in the plane using, e.g., transparencies and geometry software.

·         I can write functions to represent transformations.

G-CO.A.3

·         I can describe the rotations and/or reflections that carry it onto itself given a rectangle, parallelogram, trapezoid, or regular polygon.

G-CO.A.4

·         I can recall definitions of angles, circles, perpendicular and parallel lines and line segments.

·         I can develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

G-CO.A.5

·         I can, given a geometric figure and a rotation, reflections or translation, draw the transformed figure using, e.g. graph paper, tracing paper or geometry software.

·         I can a draw transformed figure and specify the sequence of transformations that were used to carry the given figure onto the other.

G-CO.B.6

·         I can use geometric descriptions of rigid motions to transform figures.

·         I can predict the effect of a given rigid motion on a given figure.

·         I can define congruence in terms of rigid motions (i.e. two figures are congruent if there exists a rigid motion, or composition of rigid motions, that can take one figure to the second).

·         I can describe rigid motion transformations.

·         I can predict the effect of a given rigid motion.

·         I can decide if two figures are congruent in terms of rigid motions (it is not necessary to find the precise transformation(s) that took one figure to a second, only to understand that such a transformation or composition exists).

·         I can, given two figures, use the definition of congruence in terms of rigid motion to decide if they are congruent.

G-CO.B.7

·         I can identify corresponding angles and sides of two triangles.

·         I can identify corresponding pairs of angles and sides of congruent triangles after rigid motions.

·         I can use the definition of congruence in terms of rigid motions to show that two triangles are congruent if corresponding pairs of sides and corresponding pairs of angles are congruent.

·         I can use the definition of congruence in terms of rigid motions to show that if the corresponding pairs of sides and corresponding pairs of angles of two triangles are congruent then the two triangles are congruent.

·         I can justify congruency of two triangles using transformations.

G-CO.B.8

·         I can informally use rigid motions to make angles and segments (from 8th grade).

·         I can formally use dynamic geometry software or straightedge and compass to take angles to angles and segments to segments.

·         I can identify ASA, SAS, and SSS.

·         I can explain how the criteria for triangle congruence (ASA, SAS, SSS) follows from the definition of congruence in terms of rigid motions (i.e. if two angles and the included side of one triangle are transformed by the same rigid motion(s) then the triangle image will be congruent to the original triangle).

G-CO.C.9

·         I can identify and use properties of vertical angles.

·         I can identify and use properties of parallel lines with transversals, corresponding angles, and alternate interior and exterior angles.

·         I can identify and use properties of perpendicular bisector.

·         I can identify and use properties of equidistant from endpoint.

·         I can identify and use properties of all angle relationships.

·         I can prove vertical angles are congruent.

·         I can prove corresponding angles are congruent when two parallel lines are cut by a transversal and converse.

·         I can prove alternate interior angles are congruent when two parallel lines are cut by a transversal and converse.

·         I can prove points are on a perpendicular bisector of a line segment are exactly equidistant from the segments endpoint.

G-CO.C.10

·         I can identify the hypothesis and conclusion of a triangle sum theorem.

·         I can identify the hypothesis and conclusion of a base angle of isosceles triangles.

·         I can identify the hypothesis and conclusion of midsegment theorem.

·         I can identify the hypothesis and conclusion of points of concurrency.

·         I can design an argument to prove theorems about triangles.

·         I can analyze components of the theorem.

·         I can prove theorems about triangles.

G-CO.C.11

·         I can classify types of quadrilaterals.

·         I can explain theorems for various parallelograms involving opposite sides and angles and relate to figure.

·         I can explain theorems for various parallelograms involving diagonals and relate to figure.

·         I can use the principle that corresponding parts of congruent triangles are congruent to solve problems.

·         I can use properties of special quadrilaterals in a proof.

G-CO.D.12

·         I can explain the construction of geometric figures using a variety of tools and methods.

·         I can apply the definitions, properties and theorems about line segments, rays and angles to support geometric constructions.

·         I can apply properties and theorems about parallel and perpendicular lines to support constructions.

·         I can perform geometric constructions including: 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 given line through a point not on the line, using a variety of tools an methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

G-CO.D.13

·         I can construct an equilateral triangle, a square and a regular hexagon inscribed in a circle.

 

 

Enduring Understandings:

 

·         Since many geometric figures in the real world are not stationary, transformations provide a way for us to describe their movement.

·         Proof and congruence are not exclusive to mathematics and the logical processes with defining principles can be applied in various life experiences.

·         Definitions establish meanings and remove possible misunderstanding. Other truths are more complex and difficult to see. It is often possible to verify complex truths by reasoning from simpler ones by using deductive reasoning.

·         The geometric relationships that come from proving triangles congruent may be used to prove relationships between geometric objects.

·         Representation of geometric ideas and relationships allow multiple approaches to geometric problems and connect geometric interpretations to other contexts.

·         Communicating mathematically appropriate arguments are central to the study of mathematics.

·         Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations.

·         Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial reasoning.

 

Essential Questions:

 

·         What is the significance of symbols and “good definitions” in geometry?

·         What are the undefined building blocks of geometry and how are they used?

·         What is nature’s geometry? How can man use nature’s geometry to improve his environment?

·         How are geometric transformations represented as functional relationships?

·         How can transformations determine whether figures are congruent?

·         How can special segments of a triangle be used to solve real-world problems?

·         How does geometry explain or describe the structure of our world?

·         How can points of concurrency be used in real-world situations?

·         How can reasoning be used to establish or refute conjectures?

·         How does my understanding of algebraic principles help me solve geometric problems?