Geometric Wonder Children

A triangle is any polygon with 3 sides and 3 angles.

Angles must add up to 180º

65 65

50

What if one angle was perpendicular, aka, 90º?

But something happens…

90 α

β

Then that means the others have to measure to 90º as well.

30+60=90

90 60

30

Hypotenuse is always opposite the R. Angle.

HypotenuseSide/Height

Side/Base

There are different kinds of right triangles:

Scalene/30-60-90 Right isosceles/45-45-90 Scalene

One 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

a

b

ca2

b2

c2

There is some consistency with angles and sides

Once you know two sides, you can figure out the third

32+42=x2

9+16=x2

25=x2

5=x

x

3

4

30-60-90

Ratios are the same for all lengths

45-45-90 Ratios are the same for

all lengths

Note when the angle is the same…

… The lengths of the sides have the same ratios!

60

302

1

√3

30

60

4

2

2√3

45

45

1.5

1.51.5√2 2

2

45

452√2

Coincidence..? I think not…

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!!

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!

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=c2

Sine

Cosine

Tangent

SohCahToa

Pythagorean Triples

"Without geometry life is pointless.” -Anonymous

Powerpoint Auto Shapes

Lang, S. & Murrow, G (1983). Geometry: a high school course. New York: Springer-Verlag.