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Geometry 1A Name _____________________ 2.5 Justifying Statements and 2-Column Proofs Perpendicular Lines Perpendicular lines are two lines that intersect to form right angles. The symbol for perpendicular to is . Two-Column Proofs A two-column proof organizes the statements and reasons used to justify the steps in two columns with statements on the left and reasons (definitions, properties, postulates, theorems) on the right. Steps for Writing a Proof 1. Rewrite the given statement in statement one. 2. Mark the picture with the given information. 3. Look at one piece of given information: a. What does it tell us? b. What can we say? c. Mark the picture with the new information 4. Look at the next piece of given information: a. What does it tell us? b. What can we say? c. Mark the picture with the new information Example 1: Use the figure to write the definition, property, postulate or theorem that justifies each statement. C Given: 2 is supplementary to BF bisects GBE HD CF CBE 1 E a. CB BF CF c. m 2 m CBE e. 180o DBF is a right angle B 6 D b. 1 5 d. 2 3 f. m FBG m GBA m FBA 2 3 F 5 A H 4 G Example 2: Write and justify two statements based on the information in the figure. Be sure to use either definitions, properties, postulates, or theorems to justify your statements. C Example 3: Complete the following proof. A B D P Q R Given: Q is the midpoint of PR QR RS Prove: PR S RS Statement s Reasons 1. Q is the midpoint of PR QR RS 2. PQ QR 1. 3. PR 3. RS 2. Example 4: Complete the proof. 1 and 2 are supplementary m 1 68o Prove: m 2 112o Given: 1 Statement s 1. 1 and 2 are supplementary m 1 68o Reasons 1. 2. 2. 3. 3. 4. 4. 2 Example 5: Complete the proof. Given: ABD and Prove: x = 10 1. Statements ABD and DBC are supplementary angles Reasons 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Example 6: Complete the proof. Given: RQV and Prove: x = 9 D DBC are supplementary angles o (3x + 45)o (12x – 15) B A R S Q (4x + 30)o (2x + 48)o SQT are vertical angles Statements 1. RQV and SQT are vertical angles 2. C Reasons 1. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. V T