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Congruence, Proof, and Constructions

Congruence, Proof, and Constructions

This chapter begins by precisely defining some geometric objects such as angles, circles, parallel lines, perpendicular lines, and line segments. Here, students need to pay attention to details to define each object correctly. In addition, except for the line segments definition, a definition in terms of transformations is presented for each of the other geometric objects. Later, transformations are defined as functions that can be applied to geometric figures. Also, it is studied when a transformation is a rigid motion as well as the composition of transformations. Having these definitions set allows the student to work with the definition of symmetry and explore whether a plane figure has some sort of symmetry - rotational symmetry or line symmetry. Finally, these two last types of symmetries are investigated through interactive applets where different transformations can be applied to a given polygon.

Next, a formal definition of rotations, translations, and reflections of plane figures are presented after having explored some properties of each of the transformations via interactive applets. Additionally, students are taught how to perform each of these transformations by hand. Students investigate the relationship between the input and output for each transformation when applied to points in the coordinate plane. Also, compositions involving either two rotations, or two translations, or two reflections, or a translation and a reflection are performed to investigate which of them can be seen as a single transformation.

The following lessons focus on developing theorems about triangle congruence, lines and angles, and parallelograms. The proofs of several of these theorems are developed using transformations. First, the definition of congruent figures in terms of rigid motions is presented. Then, congruent triangles are properly defined, and different criteria for proving triangle congruence are explored and established. The congruence theorems presented include the Side-Angle-Side Theorem, the Angle-Side-Angle Theorem, the Side-Side-Side Theorem, and the Angle-Angle-Side Theorem. Additionally, the Hypotenuse-Leg Congruence Theorem for right triangles is given. Leaving triangle congruence behind, the next lesson focuses on establishing relationships between the angles formed when a transversal cuts two parallel lines. These relationships are first investigated using interactive applets and then are formally stated and proven. The resulting theorems introduced include the Vertical Angles Theorem, the Corresponding Angles Theorem, the Alternate Interior Angles Theorem, the Alternate Exterior Angles Theorem, and the Perpendicular Bisector Theorem.

The next two lessons are fully devoted to investigating different properties of triangles, like the fact that, no matter the type of triangle, the sum of the interior angles is always 180 degrees. The definitions of centroid, circumcenter, incenter, and orthocenter are given along with a theorem highlighting their property. Relationships between the base angles of an isosceles triangle are set as well as the relationship between the midsegment of a triangle and its third side. Finally, it is mentioned that the centroid, the circumcenter, and the orthocenter of a triangle are collinear, and the line passing through them is called a Euler's Line. To close this group of theorem lessons, parallelograms are studied with the aim of discovering their different properties. Starting with general parallelograms, relationships between opposite sides, opposite angles, and diagonals are investigated using interactive applets. Additionally, criteria for determining when a quadrilateral is a parallelogram are presented. Similarly, criteria for determining whether a parallelogram is a rectangle or a rhombus are shown.

Before finishing the chapter and motivated through different applets, students learn how to perform different geometric constructions using different tools like a compass, a straightedge, a piece of string, or folding paper. The constructions include copying angles and segments, drawing parallel or perpendicular lines to a given line, drawing an angle bisector, and drawing a perpendicular bisector. The chapter finishes by showing in detail how to construct regular polygons inscribed in a circle - equilateral triangles, squares, regular pentagons, and regular hexagons. Once a definition or theorem is taught, it is put into practice by solving real-world situations.