Concepts of Congruence
Unit description: In this unit, students learn about translations, reflections, and rotations in the plane and how to use them to define the concept of congruence. They will learn to use and apply knowledge of rigid motions to determine similarity and congruence when solving real world problems. They will learn to identify a sequence of transformation that will map a figure onto itself. Students will learn to prove/disprove similarity/congruence using translations, reflections and rotations. Students will learn to use knowledge of angle pairs, degrees of a triangle and exterior angles to solve for missing angles. Students will learn to use the Pythagorean Theorem to find missing sides of a triangle, find distance in the coordinate plane, and solve real-world problems.
Essential Outcomes of the Unit
Geometry- Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.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.
Standards
Geometry: Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.1. Verify experimentally the properties of rotations, reflections, and translations.
- Verify experimentally lines are mapped to lines, and line segments to line segments of the same length.
- Verify experimentally angles are mapped to angles of the same measure.
- Verify experimentally parallel lines are mapped to parallel lines.
- 8.G.2. Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane.
Geometry: Understand and apply the Pythagorean Theorem.
- 8.G.6. Understand a proof of the Pythagorean Theorem and its converse.
- 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Essential Questions and Big Ideas
- What are the properties of basic rigid motions?
- Reflections, translations and rotations preserve congruence, size and sharpe, and therefore are called rigid transformations.
- How do we map the sequence between two congruent figures on a coordinate plane?
- By testing transformations of the original figure on the coordinate plane we can determine the sequence of transformations that creates a congruent figure.
- How can we determine congruence using angle relationships?
- Angle Sum Theorem can be used to determine congruence of angles in a triangle
- Alternate Interior Angles Theorem can be used to determine congruence of alternate interior angles of parallel lines cut by a transversal.
- How does the Pythagorean Theorem help solve real world problems?
- Pythagorean theorem to find the length of a diagonal of a rectangle.
- Pythagorean theorem to find the missing length of the side of a right triangle.
Prerequisite Standards
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
- 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
- 7.G.6 Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids.
Note: The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.” (This definition includes parallelograms and rectangles.)
Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids.
Note: Right prisms include cubes.
Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms.
Geometric measurement: understand concepts of angle and measure angles.
- 4.MD.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
- An angle that turns through 𝑛𝑛 one-degree angles is said to have an angle measure of 𝑛𝑛 degrees.
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
- 4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
- 4.G.2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
- 4.G.3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
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