A/N: Disclaimer from before still stands for all chapters--me too lazy to rewrite over and over and over…

Please I would be ever so grateful if you would review!

Chapter 3

As she hiked down the road, Stella's eyes took in the scenery around her. Slowly, the lush and dense forest gave way to a grassy plain. Long grass rustled in the cool breeze and the sky overhead was a clear, cloudless azure. Geomno is quite a lovely place, Stella thought. Her mind turned to all the events that she had experienced in the past few hours and marveled at how impossible everything seemed. Surprisingly, Stella hadn't felt the least bit anxious about finding a way home. She felt safe in Geomno, and had no doubt that all of her questions would be answered in time.

Stella's attention turned back to the scenery, as she noticed a house located a little ways off the road. It looked like a small Spanish mansion. Stella made her way over to it. "Hello?" She called out. This might be where Euclid's friend lived.

Stella heard the sound of a door opening somewhere above her head. "Buenos Dies, Senorita!" A cheerful voice called out. It was evident that Spanish was the person's native language.

Stella looked up and her eyes came across a middle-aged Spaniard with short black hair and glasses. The man wore a red jacket over a crisp white dress shirt. He was standing on a small balcony that protruded from the house, with his hands resting on the decorated iron fencing. Stella noticed that his hands were covered in ink stains, and figured he must be an artist or writer of some kind. "How may I help you?" the man asked with a warm tone.

"Hello there! Are you a friend of Euclid's?"

"Why yes, I am. My name is Santiago Calatrava. What is yours?"

"My name is Stella. It's nice to meet you Mr. Calatrava."

"You can just call me Santiago."

"Oh, thank you. I was wondering if you might be able to answer a question of mine."

"Well, I can certainly try."

"Ok! But in order for it to make sense, I think I need to tell you a little about the situation I'm in."

"That sounds reasonable. Go right ahead, Stella."

Stella proceeded to tell Santiago about how she came to Geomno and her experiences so far. When she was done, she asked her question. "Do you know how I can get back home?"

The Spaniard looked thoughtful for a moment. "I'm not quite sure. It might take me a while to formulate a theory."

"Oh." Stella couldn't hide the disappointment in her voice.

"But…while I do, would you like to come inside for a cup of chocolate?"

""Sure!" Stella smiled. She hadn't eaten or drank anything since breakfast, which now seemed to be eons in the past.

"Good. The door is over there." Santiago pointed to the left side of the house. "I'll meet you over there and let you in."

"Thank you very much." The man smiled in return before disappearing behind a curtain that draped in front of the door to the balcony.

Stella rounded the house and found Santiago waiting for her. She followed him as he led her into a warm, cozy kitchen. The Spaniard took a mug and ladled a helping of hot chocolate from a pot on the stove into it. "Here you are." He said as he handed the mug to Stella.

"Thank you." Stella took a sip. "Wow this is delicious!"

"You're welcome. That is my family's special recipe." Santiago winked. "Have a seat."

Stella pulled out a cushioned stool from the island in the middle of the kitchen and plopped down on it. Santiago switched off the stove and followed suit. "Well Stella," he said, "do you know of angles?"

"Oh, but of course!" Stella exclaimed. "An angle consists of two different rays that have the same initial point. The rays are the sides of the angle, and the initial point is the vertex of the angle."

"Muy bien!" Santiago beamed. "I am an architect, so angles are very important to my line of work. Now, angles that have the same measure are called congruent angles." Santiago whipped out a long piece of paper and a pen. "Let's say that I have an angle, A, and the measure of that angle is 90 degrees." On the paper, the Spaniard drew a right angle. "In writing, we would represent the measure of an angle by saying that the measure of angle A is equal to ninety degrees, or m A90 degrees."

"Ok." Stella took another sip of chocolate and set her mug down on the island, next to the paper.

"You also probably know that a 90 degree angle is called a right angle. It is represented by drawing a square at the angle, like this." Santiago drew a symbol on the angle.

(LEAVE SPACE FOR SYMBOL)

"Oh yea, I knew that already." Stella said.

"Then you also know that an acute angle is an angle whose measure is less than 90 degrees, and an obtuse angle is an angle whose measure is more than 90 degrees?"

"Of course. I also know that a straight angle is an angle whose measure is equal to 180 degrees, and is also known as a straight line."

"Precisely! Now, two angles who share a common vertex and a side, but have no common interior points are called adjacent angles. Interior points are points that are between each side of an angle. Exterior points don't lie in the interior of an angle."

"Alright." Stella nodded, digesting the information.

"Now I'm going to tell you about segment and angle bisectors."

"What are those?"

"Bisectors divide segments and angles into two congruent segments, or angles. The midpoint of a segment is the point that bisects the segment into two congruent segments. A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint. Compasses and straightedges (the latter being a ruler without marks) are used to construct segment bisectors. An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. Do you understand all that?"

"Yup!" Stella nodded.

"Muy Bien! Now, two angles whose sides form two pairs of opposite rays are called vertical angles." On the paper, Santiago drew an 'X' shape, numbering each angle.

(LEAVE SPACE FOR DRAWING)

"Angle one and angle three are both vertical angles."

"Doesn't that mean that angle two and angle four are also vertical angles?"

"Si! That's right. Now, linear pairs are two adjacent angles whose noncommon sides are opposite rays."

"Such as angle one and angle two? Do those two make a linear pair?"

"Yes. I see that you understand this perfectly, Stella! Congratulations!"

"Thank you."

"You're welcome. Are you familiar with complementary and supplementary angles?"

"Yes, I remember those. Two angles are complementary if the sum of their measure is 90 degrees. Two angles are supplementary if the sum of their measure is 180 degrees."

"Good! Now you must learn a few theorems and postulates, as well as their uses."

"I don't like memorizing things much."

"Memorization is a part of life, Stella! But, I do have a method to make the job a little simpler for you."

"Oh, thank you!"

"No worries. Now, let's begin…" Santiago drew a T-chart. He labeled the left column 'Statements', and the right column 'Reasons'. "This is the structure of a two-column proof. What we are going to do is actually prove statements. To do this, we first need something to prove." Santiago drew three angles, which all looked to be similar. He labeled each with a letter: A, B, or C. "There are a few facts that you will be notified of before starting a proof. In our case, we know that angle A is congruent to angle B, and angle B is congruent to angle C. We want to know whether or not angle A is congruent to angle C."

"Can't we automatically assume from that information that the answer is yes?"

"Ah! But what tells you this? We need to figure it out with logic and rules, Stella. Let me introduce you to our first theorem: The Properties of Angle Congruence, or the PAC for short."

"What does it have to do with any of this?"

"The PAC states that angle congruence is reflexive, symmetric and transitive."

"What does that mean?"

"Let me explain this to you: The reflexive part of the statement means that for any angle A, angle A will always be congruent to itself."

"But isn't that obvious?"

"Yes, but it is an important thing to know when writing proofs. Think for a moment, Stella: all of the information in your head that leads you to believe that angle A is congruent to angle C is being written down in this proof. You get to see how your brain figures out why they are congruent, but out on paper."

"Oh, I see! I'm glad Goldbach taught me about logic now."

"Yes, logic is very useful in geometry. Now, the symmetric part of the PAC means that if angle A is congruent to angle B, then angle B is congruent to angle A. Do you understand?"

"Yes-it's kind of like a mirror."

"It is exactly that, Stella. The last part of the PAC, which is the transitive, states that if angle A is congruent to angle B, and angle B is congruent to angle C, then angle A is congruent to angle C."

"Well that's it, isn't it? We've just proved that angle A is congruent to angle C through the transitive of the PAC."

"Not quite. We need to write it down in a proof. The first step to writing a proof is to state the already obvious. Just watch for a moment." Stella watched as Santiago numbered down the left column, and then wrote in angle A is congruent to angle B, angle B is congruent to angle C in the 'Statements' column. He then proceeded to number the right column, corresponding it to the left. Under the number one, he wrote in the word Given. "This is how a proof works. In the right column, you write statements. In the left column, you validate your reasons for the statement. The given will always be included in the proof."

"I see." Stella said, leaning over the paper. "What happens next?"

Santiago chuckled. "A great many things. Stating the obvious is always the first step. What is obvious in this problem, Stella?"

"Well…the measure of angle A is equal to the measure of angle B."

"Muy bien! And why is this?"

"Why-that is what a congruent angle is!"

"In a proof, it would be written as the definition of congruent angles." Santiago filled out the chart, and looked up. "What's next, Stella?"

Stella concentrated hard, her brow furrowing, looking for obvious things that could help her prove her point. "The measure of angle B is congruent to angle C?"

"What would be your reason? You must always have a reason."

"The definition of congruent angles."

"Si, Stella! And after that, what would you do?" He began filling out the third statement and reason for what she had supplied, listening for her answer to the next question.

"The measure of A is congruent to the measure of angle C? By…the transitive property?"

"The transitive property of equality. But, yes, you are right again!" He filled out the next line. "What is missing?"

"Well…oh! I see: angle A is congruent to angle C, by the definition of congruent angles."

"Wonderful! You have just completed your first proof, Stella. Congratulations!"

"Thank you!"

"I have but a few more things to introduce you to: perimeter, circumference, and area."

"Oh, but I know them already!"

"What are the formulas for perimeter and area for a square?"

"The perimeter equals four times the side length. The area is equal to the side length squared."

"How about the triangle?"

"The perimeter is equal to the sum of the side lengths, and the area is equal to one half of the base times the height."

"Muy bien, Stella! I am done teaching you now, and may I remark upon what a wonderful student you were?"

"Thank you!"

"I regret to say that I have not come up with much of an answer to your dilemma. The best thing I could say was to seek out the book that you traveled through."

"But the book is still in my Geometry classroom, which is in a different world! How can I reach it from here in Geomno?"

"Alas, but I do not know. However, can you prove that the book did not travel with you? In fact, can you prove of its location at all?"

"Well, no…"

"So you have not lost hope yet, my friend. I know of a person who lives near here, and they might be able to help you better than I could hope to. If you follow the road, you will find them."

"Oh thank you, Santiago! I am very grateful of your help. I hope I will see you again someday!"

"I have no doubt that our paths will cross again in the future. May you have luck in your search!"

"Goodbye!" Stella walked out the door and back on the path.