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# Geometry lesson

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A Lesson on Basic Geometry Proof

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### Geometry lesson

1. 1. Geometry Lesson Bob Roach ED538 Fall 2010
2. 2. Geometry Grade 9-11 California Geometry Standards Grades 8 through 12 2.0 Students write geometric proofs, including proofs by contradiction.
3. 3. Warm Up Exercise On a piece of paper to be handed in, copy each of the three statement sets then draw (write) a conclusion if possible: 1) All rectangles have congruent diagonals. ABCD is a rectangle. 2) All squares have four congruent sides. GHIJ has four congruent sides. 3) If l is perpendicular to m, then ABC is a right angle. If ABC is a right angle then BCD is complementary to BDA. l is perpendicular to m.
4. 4. Warm Up Exercise Answer (1) All rectangles have congruent diagonals. ABCD is a rectangle. We conclude that ABCD has congruent diagonals. a) All rectangles have congruent diagonals b) ABCD is a rectangle c) ABCD has congruent diagonals Statements Reasons a) Given b) Given c) Law of Detachment
5. 5. Warm Up Exercise Answer (2) All squares have four congruent sides. GHIJ has four congruent sides. a) All squares have four congruent sides b) GHIJ has four congruent sides. c) No conclusion Statements ReasonsReasons a) Given b) Given c) p → q ; q
6. 6. Warm Up Exercise Answer (3) a) l ⟂ m → ABC is a right angle b) ABC is a right angle → BCD is complementary to BDA c) l ⟂ m d) l ⟂ m → BCD is complementary to BDA e) BCD is complementary to BDA Statements Reasons a) Given b) Given c) Given d) Syllogism using conditional a and b a) p → q q → r ___________ p → r e) Law of Detachment c and d
7. 7. Five Essential Parts of a Proof State the theorem to be proved List the given information If possible draw a diagram to illustrate State what it is to be proved. Develop the proof using deductive reasoning
8. 8. What is a Theorem? Statements that are proved by using: Definitions Postulates Undefined terms (line, point, plane,...) Deductive reasoning are called Theorems We then use proved theorems to prove other theorems
9. 9. Verifying Segment Relationships Theorem 2-1: Congruence of segments is reflexive, symmetric, and transitive AB ≅ AB (Reflexive) If AB ≅ CD then CD ≅ AB (Symmetric) If AB ≅ CD and CD ≅ EF then AB ≅ EF (Transitive) Proofs: Use properties of real numbers with the definition of segment congruence (have same length)
10. 10. Proof of Theorem 2-1 (Symmetric Part) Statements Reasons a) Given b) Definition of ≅ line segments c) Symmetric property(=) d) Definition of ≅ line segments Given: PQ ≅ RS Prove: RS ≅ PQ P Q SR a) PQ ≅ RS d) RS ≅ PQ b) PQ = RS c) RS = PQ
11. 11. Example of Line Segment Proof Statements Reasons a) Given b) Definition of ≅ line segments c) Addition property(=) d) Segment addition postulate e) Substitution property (=) f) Definition of ≅ line segments Given: PQ ≅ XY QR ≅ YZ Prove: PR ≅ XZ X Z a) PQ ≅ XY QR = YZb) PQ = XY Y P RQ QR ≅ YZ c) PQ + QR = XY + YZ d) PR = PQ + QR c) PQ + QR = XY + YZ XZ = XY + YR f) PR ≅ XZ e PR = XZ
12. 12. Tears of Joy - Govi