Do Now #15:Do Now #15:

1. Find the measure of 1. Find the measure of MNMN if if NN is is between between MM and and PP, , MPMP = 6x – 2, = 6x – 2, MNMN = = 4x, and 4x, and

MP = 16.MP = 16.

2. 2. Name the postulate used to solve Name the postulate used to solve the above question.the above question.

2.4 Use Postulates 2.4 Use Postulates and Diagramsand Diagrams

Objective: To use postulates Objective: To use postulates involving points, lines and involving points, lines and

planesplanes

Postulate 6Postulate 6

A line contains at least two points.A line contains at least two points.

If Then

Postulate 7Postulate 7

If two lines intersect, then their If two lines intersect, then their intersection is exactly one point.intersection is exactly one point.

If Then

Postulate 8Postulate 8

Through any three noncollinear Through any three noncollinear points there exists exactly one points there exists exactly one plane.plane.

If Then

Postulate 9Postulate 9

A plane contains at least three A plane contains at least three noncollinear points.noncollinear points.

If Then

Postulate 10Postulate 10

If two points lie in a plane, then If two points lie in a plane, then the line containing them lies in the line containing them lies in the plane.the plane.

If Then

Postulate 11Postulate 11

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

If Then

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.

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

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

EXAMPLE 5: 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 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 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.