1. Five-Minute Check (over Lesson 2–4) Then/Now New Vocabulary
Postulates: Points, Lines, and Planes
Key Concept: Intersections of Lines and Planes
Example 1: Real-World Example: Identifying Postulates
Example 2: Analyze Statements Using Postulates
Key Concept: The Proof Process
Example 3: Write a Paragraph Proof
Theorem 2.1: Midpoint Theorem
Lesson Menu

2. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are supplementary. Conclusion: mA + mB = 180
A. valid
B. invalid
5-Minute Check 1

3. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: Polygon RSTU has 4 sides. Conclusion: Polygon RSTU is a square.
A. valid
B. invalid
5-Minute Check 2

4. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are congruent. Conclusion: ΔABC exists.
A. valid
B. invalid
5-Minute Check 3

5. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are congruent. Conclusion: A and B are vertical angles.
A. valid
B. invalid
5-Minute Check 4

6. Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: mY in ΔWXY = 90. Conclusion: ΔWXY is a right triangle.
A. valid
B. invalid
5-Minute Check 5

7. How many noncollinear points define a plane?
B. 2
C. 3
D. 4
5-Minute Check 6

8. Splash Screen

9. Lesson 2-5: Postulates and Paragraph Proofs (Pg. 125)
TARGETS
Identify and use basic postulates about points, lines, and planes.
Write paragraph proofs.
Vocabulary

10. Content Standards
G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Mathematical Practices
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
6 Attend to precision.
8 Look for and express regularity in repeated reasoning.
Vocabulary

11. Analyze statements in if-then form.
Write converses, inverses, and contrapositives. (Lesson 2-3)
Identify and use basic postulates about points, lines, and planes.
Write paragraph proofs.
Then/Now

12. postulate axiom proof theorem deductive argument paragraph proof
informal proof
Vocabulary

13. Concept

14. Concept

15. Identifying Postulates
ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.
A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q.
Answer: Points F and G lie on line m, and the line lies in plane Q. Postulate 2.5, which states that if two points lie in a plane, the entire line containing the points lies in that plane, shows that this is true.
Example 1

16. B. Points A and C determine a line.
Identifying Postulates
ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true.
B. Points A and C determine a line.
Answer: Points A and C lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true.
Example 1

17. ARCHITECTURE Refer to the picture
ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. A. Plane P contains points E, B, and G.
A. Through any two points there is exactly one line.
B. A line contains at least two points.
C. A plane contains at least three noncollinear points.
D. A plane contains at least two noncollinear points.
Example 1

18. ARCHITECTURE Refer to the picture
ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. B. Line AB and line BC intersect at point B.
A. Through any two points there is exactly one line.
B. A line contains at least two points.
C. If two lines intersect, then their intersection is exactly one point.
D. If two planes intersect, then their intersection is a line.
Example 1

19. If plane T contains contains point G, then plane T contains point G.
Analyze Statements Using Postulates
A. 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.
Example 2

20. contains three noncollinear points.
Analyze Statements Using Postulates
B. 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.
Example 2

21. A. Determine whether the statement is always, sometimes, or never true
A. Determine whether the statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point.
A. always
B. sometimes
C. never
Example 2

22. B. Determine whether the statement is always, sometimes, or never true
B. Determine whether the statement is always, sometimes, or never true. 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.
A. always
B. sometimes
C. never
Example 2

23. Q: What are paragraph proofs?
Q: What is a proof?
A: A proof is a logical argument that demonstrates that some statement is in fact true.
Q: What are paragraph proofs?
A: A paragraph proof is another way a proof is often written. The advantage of a paragraph proof is that you have the chance to explain your reasoning in your own words. In a paragraph proof, the statements and their justifications are written together in a logical order in a paragraph form. There is always a diagram and a statement of the given and prove sections before the paragraph.
Concept

24. Concept

25. Given: Prove: ACD is a plane.
Write a Paragraph Proof
Given:
Prove: ACD is a plane.
Proof: and 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 Points A, C, and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line.
Example 3

26. Example 3

27. Proof: ?
Example 3

28. A. Definition of midpoint B. Segment Addition Postulate
C. Definition of congruent segments
D. Substitution
Example 3

29. Concept

30. End of the Lesson