Postulates and Paragraph Proofs: Understanding Geometry Basics

1. Lesson 2-5 Postulates and Paragraph Proofs

2. Transparency 2-5 5-Minute Check on Lesson 2-4 • Determine whether the stated conclusion is valid based on the given information. If not, write invalid. • 1. Given: A and B are supplementary. • Conclusion: mA + mB = 180. • Given: Polygon RSTU is a quadrilateral. • Conclusion: Polygon RSTU is a square. • 3. Given: ABC is isosceles. • Conclusion: ABC has at least two congruent sides. • 4. Given: A and B are congruent. • Conclusion: A and B are vertical. • 5. Given: mY in WXY is 90. • Conclusion: WXY is a right triangle. • 6. Which is a valid conclusion for the • statement R and S are vertical angles? Standardized Test Practice: A mR + mS = 180. B mR + mS = 90. R  S. D C R and S are adjacent.

3. Transparency 2-5 5-Minute Check on Lesson 2-4 • Determine whether the stated conclusion is valid based on the given information. If not, write invalid. • 1. Given: A and B are supplementary. • Conclusion: mA + mB = 180. valid • Given: Polygon RSTU is a quadrilateral. • Conclusion: Polygon RSTU is a square. invalid • 3. Given: ABC is isosceles. • Conclusion: ABC has at least two congruent sides. valid • 4. Given: A and B are congruent. • Conclusion: A and B are vertical. invalid • 5. Given: mY in WXY is 90. • Conclusion: WXY is a right triangle. valid • 6. Which is a valid conclusion for the • statement R and S are vertical angles? Standardized Test Practice: A mR + mS = 180. B mR + mS = 90. R  S. D C R and S are adjacent.

4. Objectives • Matrix Logic • Identify and use basic postulates about points, lines and planes • Write paragraph proofs

5. Vocabulary • Axiom – or a postulate, is a statement that describes a fundamental relationship between the basic terms of geometry • Postulate – accepted as true • Theorem – is a statement or conjecture that can be shown to be true • Proof – a logical argument in which each statement you make is supported by a statement that is accepted as true • Paragraph proof – (also known as an informal proof) a paragraph that explains why a conjecture for a given situation is true

6. Matrix Logic On a recent test you were given five different mineral samples to identify.You were told that: Sample C is brown Samples B and E are harder than glass Samples D and E are redUsing your knowledge of minerals (in the table below), solve the problem

7. 5 Essential Parts of a Good Proof • State the theorem or conjecture to be proven. • List the given information. • If possible, draw a diagram to illustrate the given information. • State what is to be proved. • Develop a system of deductive reasoning.Postulate 2.1 Through any two points, there is exactly one line.Postulate 2.2 Through any three points not on the same line, there is exactly one plane.Postulate 2.3A line contains at least two points.Postulate 2.4 A plane contains at least three points not on the same line.Postulate 2.5 If two points lie in a plane, then the entire line containing those points lines in the plane.Postulate 2.6 If two lines intersect, then their intersection is one point.Postulate 2.7 If two planes intersect, then their intersection is a line.

8. Theorem 2.1 Midpoint Theorem __ __ __If M is the midpoint of AB, then AMMB.

9. Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane.

10. Determine whether the following statement is always, sometimes, or never true. Explain. For , if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes.

11. contains three noncollinear points. Determine whether the following statement is always, sometimes, or never true. Explain. Answer: Never; noncollinear points do not lie on the same line by definition.

12. Determine whether each statement is always, sometimes, or never true. Explain. a. Plane A and plane B intersect in one point. b. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. Answer: Never; Postulate 2.7 states that if two planes intersect, then their intersection is a line. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line.

13. Determine whether each statement is always, sometimes, or never true. Explain. c. Two planes will always intersect a line. Answer: Sometimes; Postulate 2.7 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect.

14. Given intersecting , write a paragraph proof to show that A, C, and D determine a plane. Given: intersects Proof: must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on Therefore, points A and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line. Prove: ACD is a plane.

15. Given is the midpoint of and Xis the midpoint of write a paragraph proof to show that

16. Proof: We are given that S is the midpoint of and X is the midpoint of By the definition of midpoint, Using the definition of congruent segments, Also using the given statement and the definition of congruent segments, If thenSince S and X are midpoints, By substitution, and by definition of congruence,

17. Summary & Homework • Summary: • Use undefined terms, definitions, postulates and theorems to prove that statements and conjectures are true • Homework: • pg 91: 3, 9, 16, 19, 28