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 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.

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

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

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 .

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

Daily Homework Quiz

For use after Lesson 2.4

1. Use the diagram to write an example of Postulate 7.

ANSWER

Line m intersects line n in point F and no other point.

Daily Homework Quiz

For use after Lesson 2.4

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.

Daily Homework Quiz

For use after Lesson 2.4

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

Daily Homework Quiz

For use after Lesson 2.4

Use the diagram to determine if the statement is true or false.

ANSWER

true

4. GH lies in plane M.

Daily Homework Quiz

For use after Lesson 2.4

Use the diagram to determine if the statement is true or false.

ANSWER

false

5. ABE and HED are vertical angles.