1. 2.4 Use Postulates and Diagrams You will use postulates involving points, lines, and planes. Essential Question: How can you identify postulates illustrated by a diagram?

2. Warm-Up Exercises 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.

3. Warm-Up Exercises 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.

4. Warm-Up Exercises GUIDED PRACTICE 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

5. Warm-Up Exercises Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. 2. GUIDED PRACTICE 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

6. Warm-Up Exercises EXAMPLE 3 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.

7. Warm-Up Exercises EXAMPLE 4 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.

8. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 3. If the given information stated PW and QW are congruent, how would you indicate that in the diagram? In Questions 3 and 4, refer back to Example 3. ANSWER

9. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 4. Name a pair of supplementary angles in the diagram. Explain. ANSWER In the diagram the angle TWP and VWP form a linear pair So, angle TWP, VWP are a pair of supplementary angles.

10. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 5. In the diagram for Example 4, can you assume plane S intersects plane T at BC ? ANSWER Yes

11. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 6. Explain how you know that AB BC in Example 4. ANSWER It is given that line AB is perpendicular to plane S, therefore line AB is perpendicular to every line in the plane that intersect it at point B. So, AB BC

12. Warm-Up Exercises Daily Homework Quiz 1. Use the diagram to write an example of Postulate 7. ANSWER Line m intersects line n in point F and no other point.

13. Warm-Up Exercises Daily Homework Quiz 2. Decide whether the statement “Two planes can intersect in exactly one point R ” is true or false. If false, give a reason. ANSWER False; Postulate 11 says that if two planes intersect, their intersection is a line.

14. Warm-Up Exercises Daily Homework Quiz Use the diagram to determine if the statement is true or false. 3. The intersection of planes M and N is points B, E and H. ANSWER false

15. Warm-Up Exercises Daily Homework Quiz Use the diagram to determine if the statement is true or false. ANSWER true 4. GH lies in plane M.

16. Warm-Up Exercises Daily Homework Quiz Use the diagram to determine if the statement is true or false. ANSWER false 5. ABE and HED are vertical angles.

17. You will use postulates involving points, lines, and planes. Essential Question: How can you identify postulates illustrated by a diagram? Analyze a diagram to determine what you can assume. Verify what you can assume by determining which postulates apply. You can identify postulates from a diagram by determining which information you can assume. Do not assume congruence, perpendicularity, collinearity, midpoints, and bisectors. Determine which postulate applies.