CK-12 Basic Geometry Volume 1 Of 2

by CK-12 Foundation

CK-12's Basic Geometry FlexBook, Volumes 1 through 2, is designed to present students with geometric principles in a more graphics-oriented course. Volume 1 includes 6 chapters: Basics of Geometry, Reasoning and Proof, Parallel and Perpendicular Lines, Triangles and Congruence, Relationships with Triangles, and Polygons and Quadrilaterals.

Genres: Mathematics, Reference, Science, Nonfiction, Education, Geometry

392 pages, Kindle Edition

First published April 18, 2012

About the author

CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the "FlexBook," CK-12 intends to pioneer the generation and distribution of high quality educational content that will serve both as core text as well as provide an adaptive environment for learning through the FlexBook Platform™.

Reviews

[Manny]

Here is a striking example where the magic phrase "Think step by step" makes ChatGPT produce a very simple mathematical proof:

mannyrayner: Is it true that every triangle has at least two acute angles?

ChatGPT: Yes, it is true that every triangle has at least two acute angles. A triangle is a polygon with three sides and three angles. The three angles of a triangle add up to 180 degrees. There are four possible types of triangles based on the measures of their angles:

Acute triangle: A triangle with all three angles measuring less than 90 degrees is called an acute triangle. An acute triangle has three acute angles.

Right triangle: A triangle with one right angle (90 degrees) is called a right triangle. A right triangle has one right angle and two acute angles.

Obtuse triangle: A triangle with one angle measuring more than 90 degrees is called an obtuse triangle. An obtuse triangle has one obtuse angle and two acute angles.

Straight triangle: A triangle with all three angles measuring exactly 90 degrees is called a straight triangle. A straight triangle has three right angles.

Therefore, every triangle, regardless of its type, has at least two acute angles. This is because the three angles of a triangle add up to 180 degrees, and any angle measuring less than 90 degrees is an acute angle.

mannyrayner: Check your definition of "straight triangle".

ChatGPT: I apologize for the error in my previous response. A triangle with all three angles measuring exactly 90 degrees is not called a "straight triangle." This is a term I made up and is not used in geometry. The correct term for a triangle with all three angles measuring exactly 90 degrees is a "right triangle," or a "rectangular triangle." A right triangle has one right angle, which is an angle measuring exactly 90 degrees, and two acute angles, which are angles measuring less than 90 degrees. I apologize for any confusion my mistake may have caused.

mannyrayner: Is it possible for a triangle to have three angles measuring exactly 90 degrees? Think step by step.

ChatGPT: Yes, it is possible for a triangle to have three angles measuring exactly 90 degrees.

Step 1: A triangle is a polygon with three sides and three angles.

Step 2: The three angles of a triangle add up to 180 degrees.

Step 3: If a triangle has three angles measuring exactly 90 degrees, then the three angles of the triangle will add up to 270 degrees (3 x 90 degrees = 270 degrees).

Step 4: Since the three angles of a triangle must add up to 180 degrees, it is not possible for a triangle to have three angles measuring exactly 90 degrees.

Therefore, it is not possible for a triangle to have three angles measuring exactly 90 degrees. A triangle with three angles measuring exactly 90 degrees would have a total angle measure of 270 degrees, which is not possible for a triangle.

I apologize if my previous response led you to believe that a triangle with three angles measuring exactly 90 degrees is possible. This was a mistake on my part and I apologize for any confusion it may have caused.

mannyrayner: Revise your last response so that the initial statement is consistent with the conclusion.

ChatGPT: I apologize for any confusion my previous response may have caused. Here is a revised version that is consistent with the conclusion:

No, it is not possible for a triangle to have three angles measuring exactly 90 degrees.

Step 1: A triangle is a polygon with three sides and three angles.

Step 2: The three angles of a triangle add up to 180 degrees.

Step 3: If a triangle has three angles measuring exactly 90 degrees, then the three angles of the triangle will add up to 270 degrees (3 x 90 degrees = 270 degrees).

Step 4: Since the three angles of a triangle must add up to 180 degrees, it is not possible for a triangle to have three angles measuring exactly 90 degrees.

Therefore, it is not possible for a triangle to have three angles measuring exactly 90 degrees. A triangle with three angles measuring exactly 90 degrees would have a total angle measure of 270 degrees, which is not possible for a triangle.