In order to prove this result, we will use

similar triangles shown in the following figure.

The midpoint theorems involve parallel lines and

similar triangles.

For example, Cavanagh (2008) encouraged students to use ratio and the principle of similar triangles to measure the height of the school flagpole.

The estimated height of the tree can be calculated using knowledge of ratio and the special case of similar triangles.

The cases that do not appear in the list are either cannot occur or lead to

similar triangles.

Wikipedia's entry about the history of trigonometry is there for all to read (on the website, several reliable sources are referenced at the bottom): "Pre-Hellenic societies such as the ancient Egyptians and Babylonians lacked the concept of an angle measure, but they studied the ratios of the sides of

similar triangles and discovered some properties of these ratios.

In particular, they focused on identifying similar triangles to determine the length of the height (figure 13).

The geometric model was also important to relate this problem with contents that include triangle inequality, Cartesian system, equation of a line, and similar triangles.

The success of this method relies on the principle of similar triangles, where two triangles have the same shape but are of different sizes.

Once similar triangles have been established, to measure the height of the tree above the eye height of the person holding the ruler, compare the large triangle ABC with the large triangle formed when Student a turns the ruler at right angles and Student [ walks away from the base of the tree (as shown in Figure 1, Position 2, and simplified as triangle BCX in Figure 3); again the triangles are similar.