Download Standards Associated with the unit

Congruence, Proofs, and Constructions
Critical Area 5: 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 solve problems about triangles, quadrilaterals, and
other polygons. They apply reasoning to complete geometric constructions and explain why they work.
Congruence, Proof, and Constructions
Experiment with transformations in the plane
Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric
concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move
objects along a circular arc with a specified center through a specified
angle.
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.
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).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it
onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
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
Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of
rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their
assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to
prove other theorems.
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.
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.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions.
Make geometric constructions.
Build on prior student experience with simple constructions. Emphasize the ability to formalize and defend how these
constructions result in the desired objects. Some of these constructions are closely related to previous standards and can
be introduced in conjunction with them.
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.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Material From Math 3 – Possibly Review
Understand and apply the Pythagorean Theorem.
Discuss applications of the Pythagorean Theorem and its connections to radicals, rational exponents, and
irrational numbers.
Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and
mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
*Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a
sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that
exhibits the congruence between them.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
coordinates.
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe
a sequence that exhibits the similarity between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For
example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line,
and give an argument in terms of transversals why this is so.
Solve real-world and mathematical problem involving volume of cylinders, cones, and spheres.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and
mathematical problems.