5.5 5.6 Indirect Proof and Inequalities in One and Two Triangle.notebook 1 November 29, 2011 School Triangle Inequalities...this concept is a part of the ACT math test! Lesson 5.5/5.6 Indirect Proof and Inequalities in One (and Two) Triangles November 29/30, 2011 Check for Understanding 3108.1.13 Use proofs to further develop and deepen the understanding of the study of geometry (e.g. twocolumn, paragraph, flow, indirect, coordinate) 3108.4.11 Use the triangle inequality theorems (e.g., Exterior Angle Inequality Theorem, Hinge Theorem, SSS Inequality Theorem, Triangle Inequality Theorem) to solve problems Nov 137:56 PM Type: Indirect Proof Idea: Assume the contradiction or conclusion is false! Once you assume it is false, you will show that the assumption leads to a contradiction. This type of proof is also called a proof by contradiction. The types of proofs we have used previously have been written using direct reasoning. We began with a true hypothesis and built a logical argument to show the conclusion was true. New Type of Proof! Nov 182:39 PM Start with something we know: Prove that ∠ADB is not a straight angle. Given: AD is perpendicular to BC. Note: most indirect proofs are written in paragraph form, but to visualize we will use a two column now. 1. ∠ADB is a straight angle. 2. m∠ADB=180 o 3. AD is perpendicular to BC 4. ∠ADB is a right angle. 5. m∠ADB=90 o 6. m∠ADB cannot equal 90 o and 180 o at the same time. 7. ∠ADB is not a straight angle. 1. Assume Opposite 2. Definition of straight angle. 3. Given 4. Definition of perpendicular lines. 5. Definition of a right angle. 6. Contradiction! 7. Proof by contradiction. Statements Reasons B C D A Nov 1811:54 AM Consider this statement: "Two acute angles do not form a linear pair" Steps: 1. Identify the conjecture to be proven. 2. Assume the opposite of the conclusion is true. 3. Use direct reasoning to show that the assumption leads to a contradiction. 4. Conclude that the assumption is false and therefore the original conjecture must be true. Example for the above statement: I. Given: <1 and <2 are acute angles Prove: <1 and <2 do not form a linear pair. 2. Assume <1 and <2 form a linear pair. 3. m<1+m<2=180 (def of linear pair) Since m<1 < 90 o and m<2 < 90 o m<1 + m<2 < 180. This is a contradiction. 4. The assumption that <1 and <2 form a linear pair is false. Therefore <1 and <2 do not form a linear pair. Draw a diagram to visual! Nov 1812:01 PM Steps: 1. Identify the conjecture to be proven. 2. Assume the opposite of the conclusion. Write this assumption. 3. Use direct reasoning to show a contradiction. 4. What can you conclude? You try: Use the following statement and answer 14. "An obtuse triangle cannot have a right angle." Nov 1812:06 PM Let's work through our next objective: Apply inequalities in one triangle. Using your paper and your geometer: 1. Sketch triangle #2 using your geometer and label the vertices as A, B, and C. 2. Using the cm ruler on your geometer, measure each side of the triangle and write those measurements on your triangle. 3. Using your protractor on your geometer, measure each interior angle of your triangle and write those degrees on your triangle for each angle. 4. Write the sides of your triangle in order from largest to smallest. 5. Write the angles of your triangle in order from largest to smallest. What do you notice about the smallest side and smallest angle? Is this true for the longest side and largest angles as well?