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Once students have justiRed SSS Triangle Congruence, SAS Triangle Congruence, and ASA Triangle Congruence, many of the proofs they can now do proceed without using transformations, but throughout the teacher materials examples are included of how theorems can be proven rigorously using both transformations and triangle congruence proofs.
Students also get the opportunity to immediately apply theorems they have proven to new contexts where those theorems help them prove new results. Many of the applications students explore involve quadrilaterals. Students learn to decompose quadrilaterals into congruent triangles, and prove many relationships within the quadrilateral hierarchy, such as that any parallelogram with at least one right angle must be a rectangle, or any quadrilateral with perpendicular diagonals that bisect each other must be a rhombus. Students will use these theorems later in coordinate geometry as they use algebraic methods to prove additional results about quadrilaterals.
Note on materials: For most activities in this unit, students have access to a geometry toolkit that includes many tools that students can choose from strategically: compass and straightedge, tracing paper, colored pencils, and scissors. In some lessons, students will also need access to a ruler and protractor. When students work with quadrilaterals, instructions for making 1-inch strips cut from cardstock with evenly spaced holes are included. These strips allow students to explore dynamic relationships among sides and diagonals of quadrilaterals. Finally, there are some activities that are best done using dynamic geometry software, and these lessons indicate that digital materials are preferred.
G3 Similarity
Before starting this unit, students are familiar with dilations and similarity from work in grade 8. They have experimentally conRrmed properties of dilations, and informally justiRed that Rgures are similar by Rnding a sequence of rigid transformations and dilations that takes one Rgure onto the other. Students have primarily explored dilations on grids, where they have additional structure to help them precisely determine if two Rgures are dilations, or to draw a dilation of a Rgure precisely.
In a previous unit in this course, students used rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence, Side-Angle-Side Triangle Congruence, and Angle-Side-Angle Triangle Congruence.
In this unit, students use dilations and rigid transformations to justify the the triangle similarity theorems of Euclidean geometry: that triangles with three pairs of congruent corresponding angles and all pairs of corresponding sides in a proportional relationship
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