Section 2.4 Use Postulates and Diagrams

Objective:

• Use postulates involving points, lines and planes

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.

Postulates

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 theirintersection is a line.

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

EXAMPLE 2 Identify postulates from a diagram

Use the diagram to write examples of Postulates 9 and 10.

Postulate 9: Plane P contains atleast 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.

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

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.

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

Section 2.4 Use Postulates and Diagrams

All point shown are coplanar

AHB and DHB are a linear pair

AHF and are vertical angles

A, H, J, and D are collinear

AD and BF intersect at H

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, F, and E are collinear

BF and CE intersect

BF and CE do not intersect

BHA CJA

AD BF or m AHB=90

G

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Interpreting a Diagram

You can Assume:

You can’t assume:

EXAMPLE 3 Use given information to sketch a diagram

SOLUTION

STEP 1

Draw TV and label points T and V.

STEP 2Draw point W at the midpoint of TV Mark the congruent segments.

STEP 3Draw PQ through W.

Sketch a diagram showing TV intersecting PQ at point W, so that TW WV .

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.

Section 2.4 Perpendicular Figures

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 SCD plane TAF 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.

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

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.

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

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