Theorem 6.9: If triangles are similar then the ratios the lengths of their corresponding sides are all the same.
Proof: We show that if the angles are congruent that the ratios of the lengths of the corresponding sides are all the same.
In the illustration
Since D ≅ A, we can move D to be on top of A, by the definition of congruent. If E is not on AB, (or it's extension) then rotate ΔDEF about the angle bisector of / D, and we will be able to assume that it is. Then rotate the figure about A until BC is vertical.
Since AB and AC intersect BC , they are not vertical, and will, thus, have finite real slopes with which we can do arithmetic.
Since / AEF ≅ / B, EF || BC, by Theorem 6.2, so |EF| is also vertical.
Let us define coordinates for these points. Let
A = D = (xo, yo)
B = (x1, y1)
C = (x2, y2)
E = (x3, y3)
F = (x4, y4)
Since BC is vertical, x1 = x2, and, since EF is vertical, x3 = x4. Let
m1 be the slope of AB
m2 be the slope of AC
Then by Theorem 1.5,
But, since x1 = x2, and x3 = x4 This gives us
As required. Since A and D were an arbitrary pair of congruent angles, the same proof could be repeated for the other pairs of corresponding sides.
next theorem (6.10)