1. Section 2.4
Use Postulates and Diagrams
Objective:
Use postulates involving points, lines and planes

2. Postulates and Theorems
Postulates are statements in geometry that are so basic, they are assumed to be true without proof.
Sometimes called axioms.
Theorems are statements that were once conjectures but have since been proven to be true based on postulates, definitions, properties, or previously proven conjectures.
Both postulates and theorems are ordinarily written in conditional form.

3. Postulates

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

5. EXAMPLE 1
Identify a postulate illustrated by a diagram
State the postulate illustrated by the diagram.
P5: Through any two points there exists exactly one line.
Postulate 10: If two points lie in a plane, then the line containing them lies in the plane

6. 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.

7. EXAMPLE 2
Identify postulates from a diagram
Use the diagram to write examples of Postulates 6 and 8
P8:Line l contains at least two points R and S
Postulate 6: Through noncollinear points R, S, and W, there exists exactly one plane M

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

9. GUIDED PRACTICE
for Example 1 and 2
Use the diagram in Example 2 to write examples of Postulates 5, 6, and 7. 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

10. Use Postulates and Diagrams
Section 2.4
Use Postulates and Diagrams
Interpreting a Diagram
You can’t assume:
You can Assume:

11. EXAMPLE 3
Use given information to sketch a diagram
Sketch a diagram showing TV intersecting PQ at point W, so that TW WV .
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.

12. Perpendicular Figures:
Section 2.4
Perpendicular Figures
Perpendicular Figures:
A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is a perpendicular to every line in the plane that intersects it at that point.

13. 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.

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

15. 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.

16. 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

17. 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