Pearson Unit 1 Topic 5: Relationships Within Triangles 5-6: Indirect Proof Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (6) Use the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Vocabulary: Indirect proof—a proof involving indirect reasoning Indirect reasoning—all possibilities are considered and then all but one are proved false. The remaining possibility must be true.

Writing an Indirect Proof: 1. State as a temporary assumption the opposite (negation) of what you want to prove. 2. State facts to show that this temporary assumption leads to a contradiction. 3. Conclude that the temporary assumption must be false and that what you want to prove must be true.

Writing the 1st step of an indirect proof: Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume? A. An integer is divisible by 5. Assume temporarily that an integer is not divisible by 5. B. You do not have soccer practice today. Assume temporarily that you do have soccer practice today.

Identifying contradictions: Which two statements contradict each other? Segments can be parallel and congruent. I and II do not contradict each other. Segments can be congruent and perpendicular. II and III do not contradict each other. Segments cannot be parallel and perpendicular. Statements I and III contradict each other.

Writing an indirect proof Given: ΔABC is scalene. Prove: A, B, and C all have different measures.

Example 1: Identify the two statements that contradict each other: I. ΔPQR is equilateral. II. ΔPQR is a right triangle. III. ΔPQR is isosceles. An isosceles triangle is defined to have at least 2 congruent sides. A triangle could be equilateral and therefore can also be called isosceles. A triangle can be a right triangle and isosceles with angles of 45-45-90. But a triangle cannot be both equilateral and a right triangle. Therefore I and II contradict each other.

Example 2: Identify the two statements that contradict each other: I. m || n. II. m and n do not intersect. III. m and n are skew. Lines cannot be both parallel and skew, so I and III contradict each other.

Example 3: Write the first step of an indirect proof of the given statement: A. At least one angle of the triangle is obtuse. At least one angle of the triangle is not obtuse. B. m2 > 90 m2 < 90

Example 4: Identify the two statements that contradict each other: I. The orthocenter of ΔABC is on the triangle. This is a right triangle. II. The centroid of ΔABC is inside the triangle. This could be any type of triangle. III. ΔABC is an obtuse triangle. I and III contradict each other.

Example 5: Write an indirect proof in paragraph form: A Euclidean triangle cannot have two obtuse angles. Assume a triangle can have two obtuse angles. The 3 angles of a triangle always equals 180. An obtuse angle is > 90. But obtuse angle + obtuse angle + 3rd angle in the triangle would be more than 180. Therefore, a Euclidean triangle cannot have two obtuse angles.