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Thales' Theorem

Two triangles are similar when they have equal angles and proportional sides.

THALES’ THEOREM: If we have three parallel straight lines, a, b and c, and they cut other two ones, r and r’, then they produce proportional segments :

When two triangles have a common angle and they have parallel opposite sides, we say that they are in Thales position:
Then they are similar ones and have proportional sides.

 

To check that two right triangles are similar, it is enough if one of their acute angles is equal or two of their corresponding sides are proportional:

We can use this in real situations:
Problem: Calculate the height of Pepito’s house considering the given distances and Pepito’s height in the picture:
    
 
 
 
 
Exercises:
1.- Leticia's cat has climbed on top of a post. Leticia can see its cat reflected in a puddle. If her eyes' height is 1.44 m, how tall is the post?
 
2.- Calculate the height of the tallest tree:
 
 
 
 
 
 
 
 
 
Solutions: 1) 3.6 m ; 2) 45.8 m