 Unit 2

Congruence

Unit Description:

In this unit, students analyze two- and three-dimensional space and figures using a precise definition of congruence.  Students study congruence by experimenting with rotations, reflections, and translations of geometrical figures. Students will examine the proof of the Pythagorean Theorem and its converse.  This work then prepares students for the more complex work of understanding the effects of dilations on geometrical figures in their study of similarity in the next unit.

Standards for 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.

Louisiana Student Standards for Mathematics (LSSM)

 G – Geometry A. Understand congruence and similarity using physical models, transparencies, or geometry software. (similarity will be developed in the next Unit) 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: (Rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8) 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. *I can define and identify rotations, reflections, and translations. *I can identify corresponding sides and corresponding angles of similar figures. *I can understand prime notation to describe an image after a translation, reflection, or rotation. *I can identify center of rotation. *I can identify direction and degree of rotation. *I can identify line of reflection. *I can use physical models, transparencies, or geometry software to verify the properties of rotations, reflections, and translations (i.e. lines are taken to lines and line segments to line segments of the same length.) 8.G.A.2 Explain 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. (Rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8) *I can define congruency. *I can identify symbols for congruency. *I can apply the concept of congruency to write congruent statements. *I can reason that a 2-D figure is congruent to another if the second can be obtained by a sequence of rotation, reflections, and translation. *I can describe the sequence of rotations, reflections, translations that exhibits the congruence between 2-D figures using words. *I can justify congruence of figures using a series of transformations. 8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (Dilations will be explored in the next Unit. Rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8) *I can describe the effects of translations, rotations, and reflections on 2-D figures using words and coordinates of the vertices. *I can distinguish between a pre-image and the image (transformed image). *I can find the coordinates of a pre-image given the image and a description of the transformations. 8.G.A.5 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. *I can define and identify transversals. *I can identify angles created when a parallel line is cut by transversal (alternate interior, alternate exterior, corresponding, vertical, adjacent, etc.). *I can justify that the sum of the interior angles equals 180. (For example, arrange three copies of the same triangle so that the three angles appear to form a line). *I can justify that the exterior angles of a triangle is equal to the sum of the two remote interior angles. *I can recognize the angles formed by two parallel lines and a transversal. *I can find the measure of angles using transversals, the sum of angles in a triangle, the exterior angles of triangles. B. Understand and apply the Pythagorean Theorem. 8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse using the area of squares. Pythagorean Theorem: If you have a right triangle, then the square of the longest side = the sum of the squares of the other two sides. Converse of Pythagorean Theorem: Given a triangle, if the square of the longest side = the sum of the squares of the other two sides, then the triangle is a right triangle. *I can define key vocabulary: square root, Pythagorean Theorem, right triangle, legs a and b, hypotenuse, sides, right angle, converse, base, height, proof. *I can identify the legs and hypotenuse of a right triangle. *I can explain a proof of the Pythagorean Theorem. *I can explain a proof of the converse of the Pythagorean Theorem. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Some parts of tasks require students to use the converse of the Pythagorean Theorem.) *I can recall the Pythagorean Theorem and its converse in order to apply it to real-world and mathematical problems using 2 and 3 dimensional figures. *I can solve basic mathematical Pythagorean Theorem problems and its converse to find missing length of sides of triangles in two and three dimensions. *I can apply the Pythagorean Theorem in solving real-world problems dealing with two and three dimensional shapes. 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. *I can recall the Pythagorean Theorem and its converse and relate it to any two distinct points on a graph. *I can determine how to create a right triangle from two points on a coordinate graph. *I can use the Pythagorean Theorem to solve for the distance between the two points.

Enduring Understandings:

*Those transformations that modify an item so that it is in a different position, but still has the same shape, size, segment lengths, and angle measures are translations, reflections, and rotations and produce congruent geometric items.

*Congruent figures have the same size and shape.

*When parallel lines are cut by a transversal, corresponding angles, alternate interior angles, alternate exterior angles, and vertical angles are congruent.

*A relationship exists between angle sums and exterior angle sums of triangles, as well as angles created when parallel lines are cut by a transversal.

*The measure of the interior angles of any triangle is 180°.

*The measure of a right triangle is always 90°.

*A relationship exists between the measure of an exterior angle and the other two angles of a triangle.

*The Pythagorean Theorem can be used both algebraically and geometrically to solve problems involving right triangles

*There is a relationship between the Pythagorean Theorem and the distance formula and both can be used to find missing side lengths in a coordinate plane and real-world situation.

Essential Questions:

*How can we best show or describe the change in position of a figure?

*What are transformations and what effect do they have on a two-dimensional figure?

*How can transformations be used to determine congruency?

*Does a two-dimensional figure change dimensions when transformed?

*How can you use coordinates to describe the result of a translation, reflection, or rotation?

*What properties of a two-dimensional figure are preserved under a translation, reflection, or rotation?

*How is deductive reasoning used in algebra and geometry?

*What angle relationships are formed by a transversal?

*Why does the Pythagorean Theorem apply only to right triangles?

*What is the correlation between the Pythagorean Theorem and the distance formula?

*How can I use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle?

*How do I use the Pythagorean Theorem to find the length of the legs of a right triangle?