1. Postulates and Paragraph Proofs
Lesson 2-5
Postulates and Paragraph Proofs

2. Standardized Test Practice:
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.
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. C
R and S are adjacent. D
R  S.

3. Standardized Test Practice:
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
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. C
R and S are adjacent. D
R  S.

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. Hardness (compared to glass)
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 red Using your knowledge of minerals (in the table below), solve the problem
Mineral
Color
Hardness (compared to glass)
Biolite
Brown or black
Softer
Halite
White
Hematite
Red
Feldspar
White, pink, or green
Harder
Jaspar
red
Sample A B C D E
Biolite
Halite
Hematite
Feldspar
Jaspar

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

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

12. Determine whether each statement is always, sometimes, or never true
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
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
Prove: ACD is a plane.
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.

15. Given is the midpoint of and X is 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 then Since S and X are midpoints, By substitution, and by definition of congruence,

17. Summary & Homework Summary: Homework:
Use undefined terms, definitions, postulates and theorems to prove that statements and conjectures are true
Homework:
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