4 7 CPCTC Congruent Triangles Applications

Transcription

1 Congruent Triangles Applications Objective: Apply CPCTC and triangle congruence theorems and make inferences about real world diagrams to determine measurements of parts of triangles. 1

2 Why do architects use triangles when building bridges, roofs on houses, and other structures? Why not a quadrilateral, why not a pentagon? If there is a single most important shape in engineering, it is the triangle. Unlike a rectangle, a triangle cannot be deformed without changing the length of one of its sides or breaking one of its joints. In fact, one of the simplest ways to strengthen a rectangle is to add supports that form triangles at the rectangle's corners or across its diagonal length. A single support between two diagonal corners greatly strengthens a rectangle by turning it into two triangles. Triangulation of material adds strength by eliminating lateral movement. As you can see in the structures around you every day, triangles are extremely important in engineering and thus an important topic for us to explore in this Geometry course. So far, we have learned how to classify triangles, identify congruent triangles, and explore the relationship between transformations and triangles. Since triangles are so important, let's do some more practice to make sure we really get it! 2

3 Things to remember: Congruence statement- a mathematical statement that indicates that two polygons are congruent by listing the vertices in the order of correspondence Example: ΔRMI ΔBGD Congruence postulates and theorems- SSS Post. ASA Post. HL Thm. (for right Δs only) SAS Post. AAS Thm. CPCTC- Corresponding Parts of Congruent Triangles are Congruent If you can show that two triangles are congruent, then the corresponding parts are also congruent. M D B Example: MI GD, and R B R I G 3

4 1. 2. You may check your work with your partner. 4

5 3. 4. You may check your work with your partner. 5

6 5. With your partner, talk through and work out the following problems: 6. 6

7 7. With your partner, talk through and work out the following problems: 8. 7

8 History: The Great Pyramid of Khufu, at Giza, Egypt, is 751 feet long on each side at the base, is 450 feet high, and is composed of approximately 2 million blocks of stone, each weighing more than 2 tons. The maximum error between side lengths is less than 0.1%. some interesting links (click the globe): dimensions outer.htm lascxujnpfs djcji8ncc2c The sloping angle of its sides is 51 51' (almost 52 ). Each side is oriented with the compass points of north, south, east, and west. Each cross section of the pyramid (parallel to the base) is a square. Until the 19th century, the Great Pyramid at Giza was the tallest building in the world. At over 4500 years in age, it is the only one of the famous Seven Wonders of the Ancient World that remains standing. According to the Greek historian Herodotus, the Great Pyramid was built as a tomb for the Pharaoh Khufu. 8

9 These figures are pyramid "nets"- they each fold up to create a pyramid. This particular type of pyramid, like the Great Pyramids in Giza, has a square base. What do you notice/infer about each of the triangles? You may discuss this briefly with your partner. 9

10 Did you notice that the triangles in these pyramids are congruent? A pyramid figure can be formed by non-congruent triangles, but the structure would not be as stable. Since the pyramids have existed for thousands of years, archaeologists, scientists, and people in general have wondered about how the pyramids were constructed so well, and so precisely. Considering the tools they had available at the time, it is quite impressive that each triangular face on a single pyramid was almost exactly congruent to the other three faces! Which triangle congruence theorem might we use to determine triangle congruence between any of these given triangles? 10

11 Did you say AAS? Thanks to the Vertical Angles Theorem, we know that the third congruence piece for these two triangles is those inner vertical angles. 11

12 9. 12

13 9. Because J H (given), side JI HI (given), and JIK cong HIG (Vert. Thm.), the two triangles are cong by SAS Post. Therefore, side JK cong HG. JK = 3, so the width across approved HG = 3. 13

14 Work through this problem with your partner:

15 On a scale of 0-5, with 0 being "...whaaa?" and 5 being "I can do this in my sleep", how confident are you feeling with this congruent triangle business? 15

16 If you rated yourself 0-3, do #11a: 11a) * *Explain in detail, using terminology from this unit. If you rated yourself 3.5-5, do #11b: 11b) Make sure to include an explanation for your reasoning. (If you are in Honors, do both!) 16

17 Last question for you today (and do this one on your own) : 12. Explain your conclusion. 17

18 Take a look at the summary on your notes. We've covered everything you need to know about triangles except for proofs! Now that you've finished, you may start on your homework. It's listed online as usual :) 18