## RWA Alternate Interior Angles

### Topic

Dimensions of the Earth

### Vocabulary

- Alternate Interior Angles

### Student Exploration

#### How can we prove the earth is round? How did Eratosthenes use alternate interior angles to prove the earth is round?

- According to Euclid, two parallel lines cut by a transversal have alternate interior angles that are equal. The Sun's rays are the parallel lines. One ray, at Alexandria, touches the tip of the obelisk and extends earthward toward the tip of the shadow of the obelisk, \begin{align*}AA^{\prime\prime}\end{align*}. It is extended to \begin{align*}AB\end{align*} in Figure 2. The other ray, \begin{align*}SO\end{align*}, at Syene, goes into the well and extends abstractly to the center of the Earth, \begin{align*}O\end{align*}. The obelisk, \begin{align*}AA^\prime\end{align*}, also extends abstractly to the center of the Earth, \begin{align*}O\end{align*}; thus, the line, \begin{align*}AO\end{align*}, determined by the tip of the obelisk and the center of the Earth is a transversal cutting the two parallel rays, \begin{align*}SO\end{align*} and \begin{align*}AB\end{align*}, of the sun.
- Angles \begin{align*}(BAO)\end{align*} and \begin{align*}(SOA)\end{align*} are thus alternate interior angles in geometric configuration described above; therefore, they are equal.
- Use the length of the obelisk shadow and the height of the obelisk to determine angle \begin{align*}BAO\end{align*}; triangle \begin{align*}AA^\prime A^{\prime\prime}\end{align*} is a right triangle with the right angle at \begin{align*}A^\prime\end{align*}. Thus, we would note, \begin{align*}\tan (A^\prime AA^{\prime\prime}) = \frac{(\text{length of shadow})}{(\text{height of obelisk})}\end{align*}. Eratosthenes's measurements of these values led him to conclude that the measure of angle \begin{align*}(A^\prime AA^{\prime\prime})\end{align*} was 7 degrees and 12 minutes. Read more at ...

### Extension Investigation

- Students read the following story of Eratosthenes's that contains a very simple explanation of how he proved the earth was round. http://www.mrsciguy.com/dimen.html
- Students may work with another school in another city via Skype to reproduce Eratosthenes's experiment to prove the Earth is round.