Unit 5 Lesson 7: Proving Triangles Similar

Transcription

1 Unit 5 Lesson 7: Proving Triangles Similar This lesson gives us an understanding of the different and most efficient ways that we can prove triangles to be similar to each other. These 2 slides explain "WHY" is sufficient for triangle similarity! Use the properties of similarity transformations to establish the criterion for two triangles to be similar. When studying congruent triangles in G.O.8 we established some minimum requirements that would guarantee congruence through a single or sequence of isometric transformations. We found that SSS, SS, S, S and HL (and some special cases of SS) to be enough information to always establish congruence between two triangles. In a likewise manner, we want to find the minimum requirements in two triangles to establish similarity. The way we are going to do this is to use a single or sequence of similarity transformations that would map one triangle onto the other. 1

2 Teacher to explain blue to yellow see single arrow see Double arrows NOTE: This is the most common way to prove triangles similar as we will see in the near future...in short ngle ngle () Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. E If < <D and < <E, then Δ ~ ΔDEF. D F 2

3 The triangles shown are similar. 1) Which angles are? E D 2) Write a statement of similarity. F 3) Write a statement of proportionality. The triangles shown are similar. 1) Which angles are? 2) Write a statement of similarity. D E 3) Write a statement of proportionality. 3

4 DO NOW!! Use the diagram to complete the following: 1) ΔPQR ~ 2) PQ = QR = RP??? 3) 20 =?? 12 4)? = 18 20? 5) x = P L y x M Q 20 R 15 N 6) y = Use the properties of similarity transformations to establish the SSS criterion for two triangles to be similar. DEF because was mapped onto DEF using only similarity transformations. Thus SSS is a similarity criterion. 4

5 Side-Side-Side (SSS) Similarity Theorem: If the corresponding lengths of the sides of two triangles are proportional, then the triangles are similar. If = = PQ QR RP P then Δ ~ ΔPQR Q R Now let us see if knowing two corresponding proportional sides and the included corresponding congruent angle (SS) is enough for establishing similarity. Just as it was important in congruence the angle must be the included angle meaning that it is the one between the two sides. 5

6 Side-ngle-Side (SS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle, and the lengths of the sides including these angles is proportional, then the triangles are similar. P If < <P and = PQ PR Q R then Δ ~ ΔPQR Prove: Δ ~ ΔED 6

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8 Sometimes we prove similarity to establish new relationships about the triangle. look! Once similarity is established we know that there are three corresponding congruent angles and that the sides are proportional. This is a bit like congruence where once we had proven the triangles to be congruent we could use PT (orresponding Parts of ongruent Triangles are ongruent) to gather new relationships about the triangle. Look at what we have to prove now!! 8

9 DE Homework U5 L7 Day #1 for Wed 2/15 Homework U5 L7 Day #2 for Thursday 2/16 On My Website 9