DECIPHERING THE PROOF
A COMPREHENSIVE SOLUTIONS GUIDE USING DEDUCTIVE GEOMETRY VOLUME 1
by
Book Cover & Preview Text
The three Geometry topics covered in this book are:
1. Triangles
2. Central Symmetry
3. Angles and Parallelograms
4. Special Triangles
5. Special Quadrilaterals
Each chapter begins with a brief Summary of Essential Theorems and Definitions. Followed by one Example explaining how to use the theorems and definitions to solve a problem. The third section is Exercises, which will provide the reader an opportunity to practice applying the theorems and definitions. To gain the most from the Exercises, they should be completed before reading the solutions. The last section is the detailed Solutions of the Exercises. Each solution is broken down into “Steps” and “Reason”
There are three ways to use this book based on your geometry knowledge.
1. For most students.
a. Read your textbook explaining the proofs for a specific topic.
b. Find the Chapter in the workbook which addresses the topic.
c. Review the Summary of Essential Theorems and Definitions and the Example.
d. Solve the Exercises.
e. Compare your answer with the solution in the Solutions section.
2. If you find this too challenging at first, then after step “c” above:
a. Go to the “Solutions” section, cover the “Reason” column of the solution, and referring to the “Summary of Essential Theorems and Definitions” try to determine the “Reason” for each “Step”.
3. Once you understand the “Reason” for each “Step”. Try solving again by covering the “Step” section, and answering each Step by only looking at the Reason.
4. Once you have been able to identify the Steps and Reason with some help (#2 and #3 above), go back and try solving the problem without looking at the Solution.
For example:
There are three triangles ABC, ABD, and ABE; with M, N, P as their respective circumcenters. Prove that M, N, and P are collinear.
Step Reason
1) In triangle ABC, M is on the perpendicular bisector of . The circumcenter is at the intersection of the three perpendicular bisectors of the triangle; hence on the perpendicular bisector (∆) of .
2) In triangle ABD, N is on the perpendicular bisector of . Same Reason as Step 1, is common to all three triangles.
3) In triangle ABE, P is also on the perpendicular bisector of . Same Reason as Step 1, is common to all three triangles.
4) Therefore, M, N, and P are collinear. They are all three on the perpendicular bisector ∆ of .
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Book Details
About the Book
Deciphering the Proof is for students, parents, and new teachers who need practice solving proofs in Geometry. Specifically, where Geometry is part of the 4e curriculum in a French program, or for American students taking Geometry between Grades 8 and 10. The book shows, step-by-step, how to reason and solve Geometry problems, by writing solutions in a clear, logical, and deductive sequence. This strategy is called, “modeling.” Students learn, by imitating the method, and eliminating all the non-value adding verbiage that distract graders. By showing the core steps required to solve a problem, students avoid extraneous text, and steps, which make the solution difficult to follow, and difficult for the grader to evaluate with precision. Teachers can use the material, in class, by showing partial solutions (of the reasoning or the proof), and asking the students to complete the other part. The book should be used as a complement to a Geometry textbook. It is especially beneficial for average students with difficulties writing the solution to a problem in a logical deductive process. It is recommended to the user of the book to, first, try to solve the problems entirely, before comparing with the step-by-step solutions following each chapter.
About the Author
Mr. Guyamier has a Baccalaureate, and a Master’s Degree, in Math Education, and over 40 years of teaching Mathematics in the U.S. and Canada. After years of using different books, he started writing this material for his own students over ten years ago, and has been refining it ever since.