Obj. 10 Geometric Proof
- 1. Obj. 10 Geometric Proof The student will be able to (I can): • Prove geometric theorems by deductive reasoning
- 2. geometric proof A proof which uses geometric properties and definitions • A two-column geometric proof begins with the GGGGiiiivvvveeeennnn statement and ends with the PPPPrrrroooovvvveeee statement. • List the steps of the proof in the left column and the justifications (reasons) in the right column. • You may use definitions, postulates, and previously proven theorems as reasons. • Other types of proofs are — Paragraph proofs — Flowchart proofs
- 3. Line segments with equal lengths are congruent, and angles with equal measures are also congruent. Therefore, the reflexive, symmetric, and transitive properties of equality have corresponding pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff ccccoooonnnnggggrrrruuuueeeennnncccceeee. Reflexive Property of Congruence fig. A @ fig. A Symmetric Property of Congruence If fig. A @ fig. B, then fig. B @ fig. A. Transitive Property of Congruence If fig. A @ fig. B and fig. B @ fig. C, then fig. A @ fig. C.
- 4. GGGGiiiivvvveeeennnn:::: ÐBAC is a right angle; Ð2 @ Ð3 PPPPrrrroooovvvveeee:::: Ð1 and Ð3 are comp. 1 2 3 • • B A C SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss 1. ÐBAC is a right angle 1. Given 2. mÐBAC = 90° 2. DDDDeeeeffff.... rrrriiiigggghhhhtttt Ð _______________ 3. mÐ1111 ++++ mmmmÐ2222 ==== mmmmÐBBBBAAAACCCC _______________________ 3. Ð Add. post. 4. mÐ1 + mÐ2 = 90° 4. Subst. prop. = 5. Ð2 @ Ð3 5. Given 6. mÐ2222 ==== mmmmÐ3 _______________________ 6. Def. @ Ðs 7. mÐ1 + mÐ3 = 90° 7. SSSSuuuubbbbsssstttt.... pppprrrroooopppp.... ==== _______________ 8. Ð1111 aaaannnndddd Ð3333 aaaarrrreeee ccccoooommmmpppp.... _______________________ 8. Def. comp. Ðs
- 5. Example: GGGGiiiivvvveeeennnn Ð1 and Ð2 are supplementary, and Ð2 and Ð3 are supplementary PPPPrrrroooovvvveeee Ð1 @ Ð3 1. Ð1 and Ð2 are supp. 1. Given Ð2 and Ð3 are supp. 2. mÐ1 + mÐ2 = 180° 2. Def. supp. Ð mÐ2 + mÐ3 = 180° 3. 180° = mÐ2 + mÐ3 3. Sym. prop = 4. mÐ1+mÐ2=mÐ2+mÐ3 4. Trans. prop = 5. mÐ1 = mÐ3 5. Subtr. prop.= 6. Ð1 @ Ð3 6. Def. @ Ðs
- 6. GGGGiiiivvvveeeennnn:::: Ð2 @ Ð3 PPPPrrrroooovvvveeee:::: Ð1 and Ð3 are supplementary 1 2 3 1. Ð2 @ Ð3 1. Given 2. Ð1 and Ð2 are supp. 2. Def. linear pair 3. mÐ1 + mÐ2 = 180° 3. Def. supp. Ðs 4. mÐ2 = mÐ3 4. Def. @ Ðs 5. mÐ1 + mÐ3 = 180° 5. Subst. prop. = 6. Ð1 and Ð3 are supp. 6. Def. supp. Ðs