Chapter 5
Stella ventured on down the path. In the distance, she could see the beginnings of another wood. She sighed. Would it be safe to stay in the pains, or try to seek shelter in the woods? Stella didn't know how far away Columbus' friends might be, and if she hadn't come upon them by twilight then she would have to be prepared to weather the night alone.
Right then, Stella's eyes spied an odd figure off the path aways to her right. It appeared to be a roof, but when Stella squinted against the dim light she saw that it was, in fact, a…triangle. Or a triangular prism to be more exact. It was also huge-a model that was at least 30 feet in height. One more odd thing struck Stella: the triangle was pink. How much odder could Geomno get?
"Hello?" Stella called out.
"Why, hello there." A mild voice spoke out. Stella looked around, and her eyes settled on an old man with a long white beard. He looked very much like Euclid in the fact that he also wore a toga, but he was much older. His blue eyes sparkled with humor and wisdom. He reminds me of Gandalf, from the Lord of the Rings, Stella thought to herself.
"How may I help you, young lady?" The man asked kindly.
"My name is Stella. Are you a friend of Columbus'? I'm looking for a book…"
"It's nice to meet you Stella. My name is Pythagoras. What book are you looking for?"
"The title of it is The Geomno. Have you come across it?"
"I am afraid not. Why do you need it?"
"Without the book, I won't be able to find my way home."
"I see. Where exactly is home?"
Stella explained the details of her predicament. Pythagoras listened attentively to her story. When she was done, he mused. "I certainly know of a few places you could look for it. First, why don't we take a step back? I know of a few things I could teach you."
"Ok."
"Define a triangle."
"A triangle is a figure formed by three segments joining three noncollinear points."
"Very good. You can classify triangles by their sides and by their angles."
"How?"
"An equilateral triangle is a triangle which has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle had absolutely no congruent sides. If you wish to classify triangles by angles, there are acute triangles, which have three acute angles. Also, there are equiangular triangles, whose angles are all congruent, the right triangles, which have one right angle, and the obtuse triangles, which have one obtuse angle."
"I see."
"We must also learn the anatomy of a triangle. Their different parts all have names. Each of the three parts joining the sides of a triangle is called a vertex. The two sides sharing a common vertex are adjacent sides. Also, the sides of right triangles and isosceles triangles have special names. In right triangles, the sides that form the right angle are the lets of the right triangle. The side opposite the right angle is the hypotenuse of the triangle. In isosceles triangles, the sides that are congruent are called the legs. The third side is the base of the isosceles triangle."
"Ok." Stella said.
"Now, when the sides of a triangle are extended, other angles are formed. The three original angles are the interior angles. The angles that are adjacent to the interior angles are the exterior angles. Do you understand?"
"Yes, I do."
"Good! Now we must learn a few theorems and postulates."
"Does all of geometry revolve around those?"
"I'd say it was the other way around. Now, the Third Angle Theorem states that if two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent. This is one method of proving that triangles are congruent. The Side Side Side Congruence Postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. The Side Angle Side Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. The Angle Side Angle Congruence Postulate states that if two angles and a non-included side of one triangle are congruent to the two angles and the corresponding noncongruent side of a second triangle, then the two triangles are congruent. The Hypotenuse-Leg Theorem states that if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent. Do you understand all of that?"
"Yes."
"Good! Now I am going to test your knowledge. Come here." Pythagoras beckoned Stella over towards a table that was, incidentally, also pink, although it was shaped in a rectangle. Situated on top of it were a few sheets of clean, white paper and a pen. Pythagoras grasped the pen, and with a few quick strokes, had drawn six triangles on the paper.
"Look at the first pair of triangles, Stella. Are they congruent?"
"Yes."
"Prove it. Give me a reason; that is what I taught you the postulates and theorems for."
"Well, all three sides on the first triangle are congruent to the tree corresponding sides of the other triangle…so I'd have to say that the postulate to prove their congruence is the Side Side Side Congruence Postulate."
"Good! That is perfectly correct. What about the second pair?"
"They are congruent by the Angle Side Angle Congruence Postulate."
"And the third?"
"It can be proven congruent by the Hypotenuse-Leg Congruence Theorem."
"Very good! You have learned all that I have to teach you. Congratulations."
"Thank you. Can you answer my question now?"
"Yes and no. I am afraid I do not know the exact whereabouts of the book you are seeking. I do, however, have a friend who might. They live over yonder in the forest, and will be able to help you." Pythagoras glanced at the darkening sky and produced a lantern from under the table. "You had better take this. The forest can be very dark, even before the sun goes down." The lantern suddenly began to glow with a warm light. He handed it to Stella. "Good luck on your quest, Stella."
"Thank you very much! Goodbye, Pythagoras!" Stella waved as she backed up towards the road.
"Goodbye, Stella."
Stella turned her back on Pythagoras and towards the direction of the woods.
