Transversals. What is a proof? A proof is logical argument in which each statement you make is backed up by a statement that is accepted true.

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1 Chapter 2: Angles, Parallel Lines and Transversals Lesson 2.1: Writing a Proof Getting Ready: Your math teacher asked you to solve the equation: 4x 3 = 2x What is a proof? A proof is logical argument in which each statement you make is backed up by a statement that is accepted true. Types of Proof 1. Paragraph or Informal Proof 2. Two-Column Proof or Formal Proof 3. Flow-Chart Proof Paragraph or Informal Proof In this type of proof, you write a paragraph to explain why a conjecture for a given situation is true. Two-Column Proof or Formal Proof This type is the formal type of proof, with statements and reasons, clearly identified and separated into columns. Flow-Chart Proof In this type of proof, it organizes a series of statements in a logical order, starting with the given statements. Each statement together with its reason is written in a box, and arrows are used to show how each statement leads to another. Flow charts can make your logic visible and can help others follow your reasoning. Structure of a Two-Column Proof 1. Statements 2. Reasons Statements It is the left column of the formal proof. Here, the kind of reasoning we used in the informal proof still holds true. But the statements in this case are all arranged in one column under the heading STATEMENTS. Reasons It is the right column of the formal proof. Here, opposite the statements are their corresponding reasons, which are also arranged in one column under the heading, REASONS. NOTE: The reason for each logical statement is based on established definitions, postulates, theorems, corollaries, and properties.

2 Did you know? Theorem It is any mathematical statement that is true and has a proof. Lemma It is usually used to prove a theorem. It is true and has a proof. We refer to it in the proof of a theorem. Corollary It is a special case of a theorem, i.e., it follows directly from a theorem. Proposition It is usually weaker than a theorem. Axiom It is a mathematical statement that is not yet proven but accepted as true. It is a mathematical assertion that is (accepted) true and has no proof, i.e., underlying assumption or building block. Example: If A = 4 and B = 1, then A + B = B + A. (Verify this on the board) The Different Parts of a Formal Proof or a Two-Column Proof are Given Below: 1. The statement of the problem if it is already in the if-then form or restatement if not. 2. The drawing or the figure which will be considered in the proof. 3. The given or the hypothesis which is the second part in the general form of the deductive structure. 4. The terms to be proven or the conclusion which is the third part in the general form of the deductive structure. 5. The formal proof which is made up of two columns, one for the statements and the other for reasons. NOTE: With regards to the sources of reasons mentioned, it is necessary that one must be able to express these into an if-then form using mathematical symbols. Here are some of the common definitions, postulates, and properties expressed in if-then form that you have studied during your Plane Geometry class. TIP: Planning a proof can be a difficult task, especially the first time you do it. One excellent planning is "thinking backward". If you know where you are headed but are not sure where to begin, begin at the end of the problem and work your way one step at a time back to the beginning.

3 Table 2.1. The List of Definitions, Postulates, and Properties in Plane Geometry

4 Working with Examples Example 1: Given: BE = ST Prove: BS = ET Example 2: Given: 1 and 2 are supplementary angles 3 and 4 are supplementary angles 2 congruent to 3 Prove: 1 is congruent to 4 using a flowchart proof Example 3: Given: 1 and 2 are right angles Prove: 1 is congruent to 2 Note: Use a Two-Column Proof and using Informal Proof Example 4: Given: 1 and 2 are complementary Prove: AT AM Example 5: Given: Segments CE and RE sharing the common endpoint E such that LE = EA and CL = RA Prove: CE = ER Example 6: Given: m SEO = m CER Prove: m 1 = m 2

5 Example 7: The Angle Bisector Theorem: If IR bisects MIT, then m MIR = m MIT 2 m MIT and m RIT = Given: IR bisects MIT Prove: m MIR = 1 2 m MIT m RIT = 1 2 m MIT Let's Practice: Direction: Prove the following. 1. Given: MI = GO Prove: MG IO 2. Given: x = 5 Prove: x = 4 3. Given: m EAL = 90 m 1 = 90 Prove: m 2 + m 3 = m 1