1) Special right triangles have specific angle measurements (30-60-90 or 45-45-90) that result in consistent side length ratios. 2) The Pythagorean theorem, a2 + b2 = c2, always applies to right triangles and relates the sides. 3) Key properties of 30-60-90 and 45-45-90 triangles include specific ratios between short, medium, and long sides that remain consistent regardless of the triangle's size.

1. Special Right TrianglesGeometric Wonder Children

2. TrianglesA triangle is any polygon with 3 sides and 3 angles.Angles must add up to 180º506565

3. β90αBut something happens…What if one angle was perpendicular, aka, 90º?

4. Then that means the others have to measure to 90º as well. 30+60=90… Wait a minute…309060

5. Hypotenuse is always opposite the R. Angle.Some DefinitionsHypotenuseSide/HeightSide/Base

6. So specialThere are different kinds of right triangles:Scalene/30-60-90 Right isosceles/45-45-90Scalene

7. PythagorasOne really smart dude, Pythagoras, studied really hard.Found this pretty fundamental theorem:Adding the squares of each side-length of a right triangle will equal the square of the hypotenuse.Or: a2+b2=c2

8. This what that looks like:c2ca2abb2

9. There is some consistency with angles and sidesOnce you know two sides, you can figure out the third32+42=x2 9+16=x2 25=x2 5=xWhy is this great?x34

10. Special Right Triangles30-60-90Ratios are the same for all lengths45-45-90Ratios are the same for all lengths

11. 30-60-90Note when the angle is the same…… The lengths of the sides have the same ratios!303024√32√3601602

12. Same is true for 45-45-90!452√2451.5√221.545451.52Coincidence..? I think not…

13. For any triangle whose angles are 30-60-90:The shortest side will be half of the length of the hypotenuse and the second longest side will equal to the length of the shortest side times the square root of 3. THIS IS ALWAYS TRUE FOR A 30-60-90 Δs!!Let’s generalize this:

14. For any triangle with 45-45-90 angles:The length of the hypotenuse will be equal to the length of either side times the square root of 2.THIS IS ALWAYS TRUE FOR 45-45-90 Δs!Similar for 45-45-90:

15. Right triangles have one fixed 90º angle; the other two angle have to equal 90-x and x, respectively.Ratios of 30-60-90 and 45-45-90 R. triangles are constant.In right triangles, Pythagoras’ theorem is always true: a2+b2=c2What we’ve learned:

16. SineCosineTangentSohCahToaPythagorean TriplesNext week:"Without geometry life is pointless.” -Anonymous

17. Powerpoint Auto ShapesLang, S. & Murrow, G (1983). Geometry: a high school course. New York: Springer-Verlag.References