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Special Right Triangles, Nets, Complementary and Supplementary Angles, and Dilations Special Right Triangles There are two special right triangles! We will use the Pythagorean Theorem to discover the relationships between the sides of the two special triangles. Isosceles Right Triangle Conjecture or 45-45-90 Rule In an isosceles right triangle, ifs the legs have 2 length s, then the hypotenuse has length s_______ s 2 Think: side – side – side 2 s 5 2 Lets try a few: Find the missing sides 1) 52= x y 52 2 2) 5y 5 2 5 2 5 x 2 52 16 16 x 2 2 16 2 8 2 2 2 The other special triangle is a 30 60 90 triangle If you fold an equilateral triangle along one of its altitudes you get a 30-60-90 triangle. Therefore, a 30-60-90 triangle is one half an equilateral triangle so it appears in math and engineering frequently as well. Side across from 30o is the shortest side, AIMS reference calls this side ____ 60 Side across from 90o is the hypotenuse 30 Side across from 60o is the medium side 30-60-90 Triangle Conjecture In a 30-60-90 triangle, (easy as 1, 2, 3) if the shorter side has length s, (think 1s) then the hypotenuse has length 2s _____and the longer leg has length s 3 ______ 60 Think: side – side 3 – 2 · side s 2s s 3 30 examples: 4. 5. y y 7.5 24 3 30 12 x 30 x 12 3 6. x 10 6 10 3 15 7.5 2 7. 8. 18 y 60 x 10 36 45 9. 7 3 60 y x 21 14 3 x 5 2 Complement and Supplement complementary angles A pair of has a sum of 90°. 1 2 90 B 20° 2 1 A B 90 70° A A pair of supplementary angles has a sum of 180°. C D 180 3 4 180 C 30° 4 3 150° D Warm-Up: Dilations Where have you heard the word “dilate” before? Eyes – more light, pupils get smaller What does it mean? To make wider or larger; cause to expand 1. Dilations Dilation: A non-rigid transformation in which the pre-image and the image are SIMILAR Dilations preserve angle measure, orientation, and collinearity Side length changes Nets The two-dimensional representation of all the faces of a 3-dimensional figure What a 3-D figure would look like if you “unfold it” Types of Nets Triangular Prism Square Prism Square Pyramid Triangular Pyramid Types of Triangles