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# Alternate Interior Angles

## Angles on opposite sides of a transversal, but inside the lines it intersects.

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Alternate Interior Angles
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## 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, AA\begin{align*}AA^{\prime\prime}\end{align*}. It is extended to AB\begin{align*}AB\end{align*} in Figure 2. The other ray, SO\begin{align*}SO\end{align*}, at Syene, goes into the well and extends abstractly to the center of the Earth, O\begin{align*}O\end{align*}. The obelisk, AA\begin{align*}AA^\prime\end{align*}, also extends abstractly to the center of the Earth, O\begin{align*}O\end{align*}; thus, the line, AO\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, SO\begin{align*}SO\end{align*} and AB\begin{align*}AB\end{align*}, of the sun.
• Angles (BAO)\begin{align*}(BAO)\end{align*} and (SOA)\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 BAO\begin{align*}BAO\end{align*}; triangle AAA\begin{align*}AA^\prime A^{\prime\prime}\end{align*} is a right triangle with the right angle at A\begin{align*}A^\prime\end{align*}. Thus, we would note, tan(AAA)=(length of shadow)(height of obelisk)\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 (AAA)\begin{align*}(A^\prime AA^{\prime\prime})\end{align*} was 7 degrees and 12 minutes. Read more at ...
http://www.personal.umich.edu/~copyrght/image/books/Spatial%20Synthesis/Eratosthenes/index.html
http://travel.sulekha.com/unfinished-and-finished-obelisks-of-aswan-egypt_egypt-travelogue-3185.htm

### Extension Investigation

1. 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
2. Students may work with another school in another city via Skype to reproduce Eratosthenes's experiment to prove the Earth is round.

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