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Geometry: 5 Triangle Theorems

Geometry: 5 Triangle Theorems. MathScience Innovation Center B. Davis. What are we studying?. 3 kinds of special triangles right triangles equilateral triangles isosceles triangles and their special properties. Why should we study these triangles and their properties?.

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Geometry: 5 Triangle Theorems

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  1. Geometry: 5 Triangle Theorems MathScience Innovation Center B. Davis

  2. What are we studying? • 3 kinds of special triangles • right triangles • equilateral triangles • isosceles triangles • and their special properties Congruent Triangles B. Davis MathScience Innovation Center

  3. Why should we study these triangles and their properties? • They are everywhere in the world around us!!! • Artists, architects, and many others use them to create designs ,buildings, and items that we see and use everyday. Congruent Triangles B. Davis MathScience Innovation Center

  4. Like what?? National Gallery of Art Washington, DC Photo by B. Davis 1995 Congruent Triangles B. Davis MathScience Innovation Center

  5. Like what?? National Gallery of Art Washington, DC NGA post card Congruent Triangles B. Davis MathScience Innovation Center

  6. Like what?? National Gallery of Art Washington, DC NGA Congruent Triangles B. Davis MathScience Innovation Center

  7. Like what?? What special triangles are these? On the left: a right triangle On the right: an isosceles triangle National Gallery of Art Washington, DC NGA Congruent Triangles B. Davis MathScience Innovation Center

  8. It’s cool to hold 19 degrees in your hands National Gallery of Art Washington, DC Photo by M. Davis 1995 Congruent Triangles B. Davis MathScience Innovation Center

  9. East Wing National Gallery of Art Washington, DC Photos by B. Davis 1995 Congruent Triangles B. Davis MathScience Innovation Center

  10. Like what?? If I hold 19 degrees at the purple dot, then how many degrees is at the orange dot? National Gallery of Art Washington, DC NGA Congruent Triangles B. Davis MathScience Innovation Center

  11. The East Wing of the National Gallery of Art was designed by Ieoh Ming Pei http://www.artcyclopedia.com/artists/pei_im.HTML Congruent Triangles B. Davis MathScience Innovation Center

  12. Ieoh Ming Pei • born in Canton, China in 1917. He left China when he was eighteen to study architecture at MIT and Harvard. • Pei worked as an instructor and then as an assistant professor at Harvard • Pei generally designs sophisticated glass clad buildings. He frequently works on a large scale and is renowned for his sharp, geometric designs. http://www.artcyclopedia.com/artists/pei_im.HTML Congruent Triangles B. Davis MathScience Innovation Center

  13. Ceiling National Science Foundation Washington, DC Photo by B. Davis 1995 Congruent Triangles B. Davis MathScience Innovation Center

  14. Wigwam Cody,Wyoming Photo by L. Campbell 1992 Congruent Triangles B. Davis MathScience Innovation Center

  15. US Capitol Washington, DC Photo by B. Davis 1995 Congruent Triangles B. Davis MathScience Innovation Center

  16. Window Yellowstone National Park Congruent Triangles B. Davis MathScience Innovation Center Photo by B. Davis 1994

  17. Interior by Horace Pippin Any triangles here? Congruent Triangles B. Davis MathScience Innovation Center National Gallery of Art

  18. Interior by Horace Pippin Notice the artist used triangles in the quilted rug. Congruent Triangles B. Davis MathScience Innovation Center National Gallery of Art

  19. Interior by Horace Pippin There are more groups of 3 I can see at least 8 more groups of 3. Can you? Congruent Triangles B. Davis MathScience Innovation Center National Gallery of Art

  20. Who was Horace Pippinwho painted Interior ? Lived 1888 - 1946 Painted Interior in 1944 http://artarchives.si.edu/guides/afriamer/pippin.htm Congruent Triangles B. Davis MathScience Innovation Center

  21. Horace Pippin http://artarchives.si.edu/guides/afriamer/pippin.htm African-American artist Horace Pippin, was injured by a German sniper during World War I. Pippin was a member of the 369th Army Regiment, the first African-American soldiers to fight overseas for the United States. Pippin's injury left him with a shattered right shoulder. Doctors attached his upper arm with a steel plate, and after healing, Horace could never lift his right hand above shoulder level. Congruent Triangles B. Davis MathScience Innovation Center

  22. ...So art and architecture are some reasons to study these special triangles.What do we need to learn??? Congruent Triangles B. Davis MathScience Innovation Center

  23. 5 Key Ideas: Here is the First • Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. 20 20 40o 40o x Congruent Triangles B. Davis MathScience Innovation Center

  24. 5 Key Ideas Two are: • Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse of the Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. 100 x 100 30o 30o Congruent Triangles B. Davis MathScience Innovation Center

  25. 5 Key Ideas Number 3 is: • Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. • Converse of the Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. If a triangle is equilateral, then it is equiangular. 60o 60o 60o Congruent Triangles B. Davis MathScience Innovation Center

  26. 5 Key Ideas and the 4th is: • Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. • Converse of the Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. • If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. 60o 60o 60o Congruent Triangles B. Davis MathScience Innovation Center

  27. 5 Key Ideas and the 5th and final idea is: • Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent. • Converse of the Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. • If a triangle is equilateral, then it is equiangular. • If a triangle is equiangular, then it is equilateral. If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  28. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  29. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  30. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  31. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  32. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

  33. 5 Key Ideas and the 5th and final idea is: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the 2 triangles are congruent. Congruent Triangles B. Davis MathScience Innovation Center

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