Students will see two proofs that the sum of angles in a triangle is \(180^\circ\). The first proof is a paper-cutting proof, which is informal, but is intended to increase their confidence in the theorem. The second proof is a more formal geometric proof. It works by drawing a line which is parallel to the base, through the apex of the triangle, then making some deductions. Students will see problems where a triangle is given with a variable relating its angles by linear expressions. For example, the angles might be \(x, 2x - 30\), and \(90\). Knowing that a triangle's interior angles sum to \(180^\circ\), make an equation whose LHS is the sum of the three angles and whose RHS is \(180\). Then use algebra to solve for the value of the variable, from which all angle measures can be determined.
Conclude by giving your students these challenges:
- Magic Square 15 Puzzle by Math Equals Love
- Reversals by NRICH
- 2012 AMC 8, Problem 7
- Dime Dimensions by Pierce School: Problem / Solution
Find the total area, that of all the green squares:
Here's the solution: Look at the squares from right to left. Call their side lengths \(s,\) \(s + 8,\) and \(s + 5,\) respectively. Thus, \(s + (s + 8) + (s + 5) = 25.\) Solving for \(s,\) we find \(s = 4.\) The area of all the squares must be \(s^2 + (s + 8)^2 + (s + 5)^2 = 4^2 + 12^2 + 9^2.\) Finally, evaluating the right-hand side gives us \(16 + 144 + 81 = 160 + 81 = 241.\)