2-68.  HOW MUCH IS ENOUGH? Ario thinks to himself, “There must be an easier way than measuring all three of the angles and all three of the sides to determine if two triangles are similar.  What if I only know that all of the angles are congruent?  Would that mean that the triangles are similar?” Before experimenting, make a prediction.  Do you think that two triangles must be similar if all three pairs of corresponding angles are congruent?  Write down your prediction and share it with your teammates. Describe a sequence of transformations to show that two triangles that have the same three angle measures are similar.  tinyurl.com/math2-268c Reflection, Translation, Dilation With partner – use transformations only Tinyurl.com/2-68bcpm for part b

2.3.1 Conditions for Triangle Similarity HW: 2-78,79,82,83 2.3.1  Conditions for Triangle Similarity September 17, 2018

Objectives CO: SWBAT determine if triangles are similar. LO: SWBAT explain how they know that the triangles are similar.

2-69. Scott is looking at the triangles below 2-69.  Scott is looking at the triangles below.  He thinks that ΔEFG ~ ΔHIJ but he is not sure that the shapes are drawn to scale. Are the corresponding angle measures all equal?  Convince Scott that these triangles are similar. m<I = 25 & m<E = 68, so yes! How many pairs of angles need to be congruent to be sure that triangles are similar?  Why?  How could you abbreviate this similarity condition? Two pairs because the third has to be the same since it has to add to 180. AA~ Teams

2-70.  After investigating AA triangle similarity, Carlos asked, “What if we only know that one pair of angles is congruent, but the two pairs of corresponding sides that make the angle are proportional?  Does that mean the triangles are similar?” Test triangles with two pairs of corresponding sides that are proportional, with the included angles congruent. Create a triangle with side lengths 4 cm and 5 cm and an angle of 20° between these two sides, as shown at right.  If a second triangle has an angle of 20°, and the two sides of the angle have lengths proportional to 4 cm and 5 cm (such as 8 cm and 10 cm), is the second triangle always similar to the first triangle?  That is, is it possible to make a second triangle with two side lengths proportional to 4 cm and 5 cm and an included angle of 20° that is not similar? tinyurl.com/math2-270a Is it possible to make a second triangle with two of the sides proportional to the given side lengths and with the same included angle that is not similar? Practice with the tinyurl.com/math2-270b Based on your investigations from parts (a) and (b), what conjecture can you make about two triangles that have two pairs of corresponding sides that are proportional and the included angles are congruent? Justify your response with transformations. They will be similar because the shape could be dilated out of the angle that is given. How can you abbreviate this triangle similarity condition? SAS~ With partner – use transformations only First translate one triangle to align the congruent angle vertices, then rotate to align the sides of the angle.  Finally, dilate from the vertex to map one triangle onto the other. tinyurl.com/math2-270a

2-72.  Kendall wanted to investigate if he could prove two triangles were similar without knowing if any of the corresponding angles were congruent.  That, is, he wanted to test whether SSS ~ is a triangle similarity condition. Before experimenting, make a prediction.  Do you think that two triangles must be similar if all three pairs of corresponding sides lengths have the same ratio?  Write down your prediction and share it with your teammates.  Experiment with Kendall’s idea.  To do this, use the eTools below to test triangles with proportional side lengths.  Begin with the side lengths listed below, then try some others.  Can you create two triangles with proportional side lengths that are not similar? Investigate, sketch your shapes, and write down your conclusion. (1/4) Triangle #1: side lengths 3, 5, 7 Triangle #2: side lengths 6, 10, 14 http://tinyurl.com/math2-72b1 (2/3) Triangle #1: side lengths 3, 4, 5 Triangle #2: side lengths 6, 8, 10 http://tinyurl.com/math2-72b2 When the sides are proportional, then the triangles are similar. (use transformations only) Partners