1. Do Now #15: 1. Find the measure of MN if N is between M and P, MP = 6x – 2, MN = 4x, and 1. Find the measure of MN if N is between M and P, MP = 6x – 2, MN = 4x, and MP = 16. MP = 16. 2. Name the postulate used to solve the above question. 2. Name the postulate used to solve the above question.

2. 2.4 Use Postulates and Diagrams Objective: To use postulates involving points, lines and planes

3. Postulate 6 A line contains at least two points. A line contains at least two points. If Then

4. Postulate 7 If two lines intersect, then their intersection is exactly one point. If two lines intersect, then their intersection is exactly one point. If Then

5. Postulate 8 Through any three noncollinear points there exists exactly one plane. Through any three noncollinear points there exists exactly one plane. IfThen

6. Postulate 9 A plane contains at least three noncollinear points. A plane contains at least three noncollinear points. If Then

7. Postulate 10 If two points lie in a plane, then the line containing them lies in the plane. If two points lie in a plane, then the line containing them lies in the plane. If Then

8. Postulate 11 If two planes intersect, then their intersection is a line. If two planes intersect, then their intersection is a line. If Then

9. EXAMPLE 1 Identify a postulate illustrated by a diagram State the postulate illustrated by the diagram. a.a. b.b. SOLUTION a.a. Postulate 7 : If two lines intersect, then their intersection is exactly one point. b.b. Postulate 11 : If two planes intersect, then their intersection is a line.

10. EXAMPLE 2 Identify postulates from a diagram Use the diagram to write examples of Postulates 9 and 10. Postulate 9 : Plane P contains at least three noncollinear points, A, B, and C. Postulate 10 : Point A and point B lie in plane P, so line n containing A and B also lies in plane P.

11. Example 3: for Example 1 and 2 Use the diagram in Example 2. Which postulate allows you to say that the intersection of plane P and plane Q is a line? 1. ANSWER Postulate 11

12. Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. 2. Example 4: for Example 1 and 2 ANSWER Postulate 5 : Line n passes through points A and B Postulate 6 : Line n contains A and B Postulate 7 :Line m and n intersect at point A

13. EXAMPLE 5: Use given information to sketch a diagram SOLUTION STEP 1 Draw TV and label points T and V. STEP 2 Draw point W at the midpoint of TV Mark the congruent segments. STEP 3 Draw PQ through W. Sketch a diagram showing TV intersecting PQ at point W, so that TW WV.

14. EXAMPLE 6: Interpret a diagram in three dimensions Which of the following statements cannot be assumed from the diagram? A, B, and F are collinear. E, B, and D are collinear. AB plane S CD plane T AF intersects BC at point B. SOLUTION No drawn line connects E, B, and D, so you cannot assume they are collinear. With no right angle marked, you cannot assume CD plane T.