We first drew a visual representation/diagram of person and persons shadow, the flagpole and the flagpoles shadow. We then added the proportions/measurements to our diagram that were necessary to calculate the height (5'4"= persons height, 9'=persons shadow, x=flagpole height, 40'=flagpole shadow). Then using the shadow method formula we plugged in the numbers and solved for x by cross multiplying (5'4"/9' = x/40'). Then we got our estimated flagpole height (24=x).

We can form 2 similar triangles with the shadow method because we justified it with AA theorem (angle angle). ∢B ≅∢E & ∢A≅∢D AA~
The first angles, both person and flagpole have a 90° angle, The second angles, both person and flagpole share the suns fixed angle. With this much information it is enough to prove that the two triangles are similar. AA theorem: 2 corresponding angles are equal 
We first assigned variables for measurements we needed to find the height of the flagpole, then drew a visual representation/ diagram of person and person to mirror, flagpole, flagpole to mirror (H=64", P=26", M=109", O=output/flagpoles height). Then using the mirror method formula we plugged in the numbers and solved for O by cross multiplying or using scale factor (CM 109"/O = 26"/64"  SF 109"/26"= O(64")).Then we got our estimated flagpole height (268.31" or 22.36').

We can form 2 similar triangles with the shadow method because we justified it with AA theorem (angle angle). ∢B ≅∢D &∢ A≅∢E AA~
The first angles, both person and flagpole have a 90° angle, The second angles, both person and flagpole share the mirrors reflection fixed angle. With this much information it is enough to prove that the two triangles are similar. AA theorem: 2 corresponding angles are equal 
We first drew and labeled a diagram representing a right angled isosceles triangle. Then using the clinometer we measured everything we needed to find the height(H=333", V=59", H+V=O= Height of flagpole). A clinometer has a straw on the on thetop and a string attached that when the tool is tilted to look through the straw and see the object it measures and gives you the angle you are creating to see the object. This tool measures the angle of elevation, or angle from the ground, in a right  angled triagnle. Then using the clinometer method formula we added the H(horizontal distance) and V(vertical distance) because when using the cinometer your looking to find the angle from your eyes because that's the only way you can measure it, but you are standing up so there is still that distance from your eyes to the floor you need to add to create a whole length as if you were laying on the floor (H+V=392"). Then for the total height of the flagpole, O. We converted the length into feet and got our estimated flagpole height (O= 392"/12 = 32.6').

Isosceles Triangle:
A triangle that has two sides of equal length and two equal angles. 