Experiment with transformations in the plane
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.
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).
3. Given a
rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop
definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
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. Understand congruence in terms of rigid motions
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.
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.
8. Explain how
the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions. Prove geometric theorems
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.
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.
10.1 Know and use the triangle inequality theorem.
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. Make geometric constructions
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.
13. Construct an
equilateral triangle, a square, and a regular hexagon inscribed in a circle.
