Unit 2 - Geometric Foundations, ConstructionS and Proofs
- Students should be provided opportunities to build a conceptual understanding of a point, line, line segment, plane, arc, and angle through modeling and exploration of real-life phenomena.
- Students should attend to precision when using definitions and symbolic notations.
- Students should be able to apply the Segment Addition Postulate and Angle Addition Postulate to solve real-life problems.
- Students should read, write, use, and interpret symbolic notation for point, line, plane, line segment, angle, circle, arc, perpendicular line, and parallel line.
- Students should have opportunities to use a variety of tools, which might include a compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.
- Student should be able to:
- Copy a segment and angle.
- Bisect a segment and angle.
- Construct perpendicular lines, including the perpendicular bisector of a line segment.
- Construct a line parallel to a given line through a point not on the line.
- Students should be given opportunities to precisely prove vertical angles are congruent.
- Students should be given opportunities to explore using visual tools in order to precisely prove when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent.
- Students should be provided with opportunities to analyze and apply theorems about lines and angles from the context of parallel lines cut by a transversal to make sense of relationships between lines and angles.
- Students should be given opportunities to precisely prove that points on the perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
- Students should be able to show and explain their reasoning used to generate their proof.
- Students should be able to apply theorems to solve problems and to prove relationships in geometric figures by applying geometric and algebraic reasoning.
- Use informal (visual) construction with tools (patty paper, protractor, etc.) to discover the angle relationships between angles formed when two lines are cut by a transversal.
- When using more than one transversal, tie into similar triangles and set up problems using triangle sum relationships (angle sum).
- Including identify alternate exterior angles, alternate interior angles, linear pairs, same side interior angles, same side exterior angles, and corresponding angles.
- Students should build on their existing understanding of the slope of a line segment developed in the Algebra: Concepts and Connections course.
- Students should be able to classify quadrilaterals as parallelograms (including rectangles, rhombi, and squares) using sides, angles, and diagonals.
- Students should be able to apply their understanding of slope of a line segment, as well as distance and midpoint formulas to classify quadrilaterals in the coordinate plane.
Textbook Connections
Module 1-2 Points, Lines, and Planes
Students analyze figures to identify points, lines, planes, and intersections of lines and planes.
Module 1-3 Line Segments
Students find measures of line segments.
Module 1-4 Distance
Students apply the Distance Formula to find the lengths of the segments.
Module 1-7 Midpoints and Bisectors
Students find midpoints and bisect line segments.
Module 2-1 Angles and Congruence
Students identify and use different types of angles.
Module 2-2 Angle Relationships
Students find measures of angles using complementary and supplementary angles and identify what can and cannot be assumed about angles in a diagram.
Module 3-4 Writing Proofs
Students analyze and construct viable arguments.
Module 3-5 Proving Segment Relationships
Students prove theorems about line segments.
Module 3-6 Proving Angle Relationships
Students prove theorems about angles.
Module 3-7 Parallel Lines and Transversals
Students identify and use relationships between parallel lines and transversals.
Module 3-9 Proving Lines Parallel
Students identify and use parallel lines by using angle relationships.
Module 3-10 Perpendiculars and Distance
Students use perpendicular lines to find distance.
Module 5-1 Angles of Triangles
Students solve problems using the Triangle Angle-Sum and Exterior Angle Theorems.
Module 5-6 Isosceles and Equilateral Triangles
Students solve problems involving isosceles and equilateral triangles using theorems of right triangle congruence.
Module 6-1 Perpendicular Bisectors
Students solve problems using perpendicular bisectors in triangles.
Module 6-2 Angle Bisectors
Students solve problems using angle bisectors.
Module 6-3 Medians and Altitudes of Triangles
Students solve problems using medians and altitudes in triangles.
Module 7-3 Tests for Parallelograms
Students prove and use the tests for parallelograms to determine whether quadrilaterals are parallelograms.
Module 7-4 Rectangles
Students recognize and apply the properties of rectangles and use them to determine whether a parallelogram is a rectangle.
Module 7-5 Rhombi and Squares
Students recognize and apply the properties of rhombi and squares.
Module 7-6 Trapezoids and Kites
Students solve problems using the properties of trapezoids and kites.
Module 1-2 Points, Lines, and Planes
Students analyze figures to identify points, lines, planes, and intersections of lines and planes.
Module 1-3 Line Segments
Students find measures of line segments.
Module 1-4 Distance
Students apply the Distance Formula to find the lengths of the segments.
Module 1-7 Midpoints and Bisectors
Students find midpoints and bisect line segments.
Module 2-1 Angles and Congruence
Students identify and use different types of angles.
Module 2-2 Angle Relationships
Students find measures of angles using complementary and supplementary angles and identify what can and cannot be assumed about angles in a diagram.
Module 3-4 Writing Proofs
Students analyze and construct viable arguments.
Module 3-5 Proving Segment Relationships
Students prove theorems about line segments.
Module 3-6 Proving Angle Relationships
Students prove theorems about angles.
Module 3-7 Parallel Lines and Transversals
Students identify and use relationships between parallel lines and transversals.
Module 3-9 Proving Lines Parallel
Students identify and use parallel lines by using angle relationships.
Module 3-10 Perpendiculars and Distance
Students use perpendicular lines to find distance.
Module 5-1 Angles of Triangles
Students solve problems using the Triangle Angle-Sum and Exterior Angle Theorems.
Module 5-6 Isosceles and Equilateral Triangles
Students solve problems involving isosceles and equilateral triangles using theorems of right triangle congruence.
Module 6-1 Perpendicular Bisectors
Students solve problems using perpendicular bisectors in triangles.
Module 6-2 Angle Bisectors
Students solve problems using angle bisectors.
Module 6-3 Medians and Altitudes of Triangles
Students solve problems using medians and altitudes in triangles.
Module 7-3 Tests for Parallelograms
Students prove and use the tests for parallelograms to determine whether quadrilaterals are parallelograms.
Module 7-4 Rectangles
Students recognize and apply the properties of rectangles and use them to determine whether a parallelogram is a rectangle.
Module 7-5 Rhombi and Squares
Students recognize and apply the properties of rhombi and squares.
Module 7-6 Trapezoids and Kites
Students solve problems using the properties of trapezoids and kites.