## Geometry Module 1, Topic G: Axiomatic Systems

Classroom Resource Information

Title:

Geometry Module 1, Topic G: Axiomatic Systems

URL:

https://www.engageny.org/resource/geometry-module-1-topic-g-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 1, Topic G, students review material covered throughout the module. Additionally, students discuss the structure of geometry as an axiomatic system.

Content Standard(s):
 Mathematics MA2015 (2016) Grade: 9-12 Geometry 4 ) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.4- Given a geometric figure of a reflection or a translation of that figure, identify if the geometric figure is a reflection or translation. Mathematics MA2015 (2016) Grade: 9-12 Geometry 8 ) Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.8- Given two congruent triangles and angle measures of one of the triangles, identify the angle measures of the other triangle. Mathematics MA2015 (2016) Grade: 9-12 Geometry 10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, 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, and the medians of a triangle meet at a point. [G-CO10] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.10- Given a measure of a leg or base angle of an isosceles triangle, identify the measure of the other leg or other base angle. Mathematics MA2015 (2016) Grade: 9-12 Geometry 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. [G-CO11] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.11- Given the measure of one side or one angle of a parallelogram, identify the measure of the opposite side or opposite angle. Mathematics MA2015 (2016) Grade: 9-12 Geometry 12 ) Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include 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-CO12] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.12- Given a drawing with angles and a protractor overlay, determine which angles are congruent. Sample image below. Image Mathematics MA2015 (2016) Grade: 9-12 Geometry 13 ) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13] Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.HS.13- Identify an equilateral triangle from a set of triangles or identify a regular hexagon from a set of hexagons. Make sure sides/angles are marked so that students can identify congruence. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 21. Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software. a. Describe transformations and compositions of transformations as functions that take points in the plane as inputs and give other points as outputs, using informal and formal notation. b. Compare transformations which preserve distance and angle measure to those that do not. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 22. Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software. a. Given a geometric figure and a rotation, reflection, or translation, draw the image of the transformed figure using graph paper, tracing paper, or geometry software. b. Specify a sequence of rotations, reflections, or translations that will carry a given figure onto another. c. Draw figures with different types of symmetries and describe their attributes. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 23. Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 24. Define congruence of two figures in terms of rigid motions (a sequence of translations, rotations, and reflections); show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other. Example: △ABC is congruent to △XYZ since a reflection followed by a translation maps △ABC onto △XYZ. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.) Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 25. Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another. a. Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. b. Verify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS). Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show that there must be a sequence of rigid motions will map one onto the other. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.) Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 30. Develop and use precise definitions of figures such as angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.30 Demonstrate perpendicular lines, parallel lines, line segments, angles, and circles by drawing, modeling, identifying or creating. Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis 31. Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats. a. Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints. b. Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180?; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity. c. Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals. Example: Prove that rectangles are parallelograms with congruent diagonals. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle. M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.
Tags: angle, arc, circle, compass, constructions, distance, equilateral triangle, geometry, hexagon, inscribe, line segment, parallel line, parallelogram, perpendicular line, polygon, proofs, reflections, rotations, square, straightedge, theorems, transformations, translations, trapezoid