1. Lesson 9

Module 2: Lesson 9

Module 2: Logic and Geometry

 

Lesson 9: Proving Angle Properties in Triangles

 
Focus

 

Martin Gardner (1914–2010) was known as the Mathematical Gamester. Gardner published a number of books of puzzles, problems, and mathematical games. One brain teaser found in his books was developed by Mel Stover of Winnipeg and is called Acute Dissection.

 

—Philip Yam. “Profile: Martin Gardner, the Mathematical Gamester (1914–2010),” Scientific American,
22 May 2010, <http://www.scientificamerican.com/article.cfm?id=profile-of-martin-gardner&page=4>;
(17 February 2011).

This photo shows a teenage boy with speech bubble with triangle and a question mark in it.

Creatas/Thinkstock


The problem asks whether it is possible to cut any triangle with one obtuse angle into smaller, acute triangles. If it cannot be done, a proof of impossibility is required. If it can be done, the solution must show the smallest number of acute triangles that an obtuse triangle can be dissected into.

 

—Martin Gardner. My Best Mathematical and Logic Puzzles. (Mineola, NY: Dover Publications, 1994)


 

To solve this problem, an understanding of angles and triangles is required. You know from previous courses that acute triangles have angles that are less than 90°. Obtuse triangles have one angle that is greater than 90°.

 

Another theorem you have been told in previous courses is that the sum of the three interior angles in a triangle is 180°. You have been given this theorem, but are you 100% sure that the sum of the interior angles in a triangle will always be 180°?

 

This lesson will help you answer the following inquiry questions:

  • How can properties of angles in triangles be proven using deductive reasoning?
  • How can the properties of angles in triangles be used to solve problems?
Assessment

All assessment items you encounter need to be placed in your course folder.

 

coursefolder

Save a copy of the Lesson 9 Assignment to your course folder. You will receive more information about how to complete the assignment later in this lesson.

 

 

Materials and Equipment
  • ruler or straightedge
  • compass
  • protractor (You may find it helpful to use Virtual Tools.)

     
    This is a screenshot for Virtual Tools.