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 Geometry Module 2: Similarity, Proof, and Trigonometry Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss...

 Students revisit what scale drawings are and discover two systematic methods of how to create them using dilations. The comparison of the two methods yield the Triangle Side Splitter Theorem and the...

 Student Outcomes Students review properties of scale drawings and are able to create them.

 Student Outcomes Students create scale drawings of polygonal figures by the Ratio Method. Given a figure and a scale drawing from the Ratio Method, students answer questions about the scale factor...

 Student Outcomes Students create scale drawings of polygonal figures by the Parallel Method. Students explain why angles are preserved in scale drawings created by the Parallel Method using the...

 Student Outcomes Students understand that the Ratio and Parallel Methods produce the same scale drawing and understand the proof of this fact. Students relate the equivalence of the Ratio and...

 Students study and prove the properties of dilations.

 Student Outcomes Students review the properties of basic rigid motions. Students understand the properties of dilations and that a dilations is also an example of a transformation of the plane.

 Student Outcomes Students prove that a dilation maps a ray to a ray, a line to a line and a circle to a circle.

 In the last lesson, students learned about the triangle side splitter theorem, which is now used to prove the dilation theorem. In Grade 8 students learned about the fundamental theorem of...

 In Grade 8, students informally showed that a dilation maps a segment to a segment on the coordinate plane. The lesson includes an opening discussion that reminds students of this fact. Next,...

 Student Outcomes Students prove that dilations map an angle to an angle with equal measure. Students understand how dilations map triangles, squares, rectangles, trapezoids, and regular polygons.

 Student Outcomes Students divide a line segment into equal pieces by the Side Splitter and Dilation Methods. Students know how to locate fractions on the number line.

 In Lesson 11, students examine the effects of dilating figures from two different centers. By experimental verification, they examine the impact on the two dilations of having two different scale...

 Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and...

 Student Outcomes Students define a similarity transformation as the composition of basic rigid motions and dilations. Students define two figures to be similar if there is a similarity transformation...

 Student Outcomes Students know the properties of a similarity transformation are determined by the transformations that compose the similarity transformation. Students are able to apply a similarity...

 Student Outcomes Students understand that similarity is reflexive, symmetric, and transitive. Students recognize that if two triangles are similar, there is a correspondence such that corresponding...

 Student Outcomes Students prove the angleangle criterion for two triangles to be similar and use it to solve triangle problems.

 Student Outcomes Students indirectly solve for measurements involving right triangles using scale factors, ratios between similar figures, and ratios within similar figures. Students use...

 Student Outcomes Students prove the sideangleside criterion for two triangles to be similar and use it to solve triangle problems. Students prove the sidesideside criterion for two triangles to...

 Student Outcomes Students state, understand, and prove the Angle Bisector Theorem. Students use the Angle Bisector Theorem to solve problems.

 The focus in Topic D is similarity within right triangles. Students examine how an altitude drawn from the vertex of a right triangle to the hypotenuse creates two similar subtriangles. Students...

 Student Outcomes Students understand that the altitude of a right triangle from the vertex of the right angle to the hypotenuse divides the triangle into two similar right triangles that are also...