Similarity

There are four types of transformations or changes that can be made to the position of a geometric figure. When a geometric figure undergoes a transformation, the line segments still result in line segments and angles still result in angles. Translations, reflections, and rotations produce congruent figures meaning the shape is identical in that the corresponding line segments maintain their same length and the corresponding angles maintain their same measure...the exact figure is simply in a new position. (CPCTC=corresponding parts of congruent triangles are congruent!) The location changes, and sometimes the orientation changes as well. Dilations produce similar figures meaning the figure is the same shape and the corresponding angles maintain their same measure but the corresponding line segments are proportional in length. (CCSS 8.G.A.4)

In CCSS 8.G.A.5 students investigate angle-angle criterion for similarity of triangles. What happened to the unmentioned third angle? Why not angle-angle-angle criterion for similarity? Ask your students. Prompt them to realize that the third angle is implied by interior angle sum of a triangle. In the lab activity below, students discover that two angle measures in a triangle do not yield congruence between the figures. This sets the stage to discuss the definition of similarity. The AngLegs Simultaneous RoundTable puts the definition to work and makes connections to dilation and percent increase. This is prerequisite knowledge for CCSS 8.EE.B.6 standard that connects slope and similar triangles including the derivation of y=mx+b.

8.G.A.5 Similarity Resources

These discovery activities highlight Common Core State Standards 8.G.A.5 and 8.EE.B.6 included in MATH-8 and Accelerated MATH-7.