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Key Concepts
- Use postulates involving points, lines and planes.
- Identify postulates from a diagram.
- Interpret a diagram in three dimensions.
Point, Line and Plane Postulates
POSTULATE 5: Through any two points, there exists exactly one line.
POSTULATE 6: A line contains at least two points.
POSTULATE 7: If two lines intersect, then their intersection is exactly one point.
POSTULATE 8: Through any three non-collinear points, there exists exactly one plane.
POSTULATE 9: A plane contains at least three non-collinear points.
POSTULATE 10: If two points lie in a plane, then the line containing them lies in the plane.
POSTULATE 11: If two planes intersect, then their intersection is a line.
Concept Summary


Perpendicular Figures
A line is a line perpendicular to a plane if and only if the line intersects the plane at a point and is perpendicular to every line in the plane that intersects it at that point.
In a diagram, a line perpendicular to a plane must be marked with a right-angle symbol.

Let’s solve some examples!
Identify a postulate illustrated by a diagram
Example 1:
State the postulate illustrated by the diagram.




Solution:
- Postulate 7: If two lines intersect, then their intersection is exactly one point.
- Postulate 11: If two planes intersect, then their intersection is a line.
Identify postulates from a diagram
Example 2:
Use the diagram to write examples of postulates 9 and 10.

Solution:
Postulate 9: Plane P contains at least three non-collinear 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.
2.3 Use the given information to sketch a diagram
Example 3:

Interpret a diagram in three dimensions
Example 4:

Questions to Solve
Question 1:
State the postulate illustrated by the diagram.

Solution:
- Postulate 5: If there are two points, there exists exactly one line passing through them.
- Postulate 9: If there is a plane, then there are at least three non-collinear points in that plane.
Question 2:
Use the pyramid to write examples of the postulate indicated.

Solution:
- Postulate 5: Only one line SZ passes through the points S and Z.
- Postulate 7: Lines SZ and ZU intersect at only one point Z.
- Postulate 9: Plane formed by the front of this pyramid (it will be called plane STU) contains at least three non-collinear points S, T and U.
- Postulate 10: Points Z and U lie in the plane STU, so the line ZU also lies in plane P.
Question 3:
Decide whether the statement is true or false. If it is false, give a real-world counterexample.
- Through any three points, there exists exactly one line.
- A point can be in more than one plane.
- Any two planes intersect.
Solution:
- False
- Counterexample: If these three points are non-collinear, then one line cannot pass through these three points.
True
False
- Counterexample: If the two planes are parallel, they will never intersect.
Key Concepts Covered
Point, line, and plane postulates
POSTULATE 5: Through any two points, there exists exactly one line.
POSTULATE 6: A line contains at least two points.
POSTULATE 7: If two lines intersect, then their intersection is exactly one point.
POSTULATE 8: Through any three noncollinear points there exists exactly one plane.
POSTULATE 9: A plane contains at least three noncollinear points.
POSTULATE 10: If two points lie in a plane, then the line containing them lies in the plane.
POSTULATE 11: If two planes intersect, then their intersection is a line
Exercise
- In triangle ABC, AD is a median. If the area of ΔABD is 15 cm sq, then find the area of ΔABC.
- ABCD is a parallelogram and BPC is a triangle with P falling on AD. If the area of parallelogram ABCD= 26 cm2, find the area of triangle BPC.
- PQRS is a parallelogram and PQT is a triangle with T falling on RS. If area of triangle
PQT = 18 cm2, then find the area of parallelogram PQRS. - ABCD is a parallelogram where E is a point on AD. Area of ΔBCE = 21 cm2. If CD = 6 cm, then find the length of AF.
- The area of triangle ABC is 15 cm sq. If ΔABC and a parallelogram ABPD are on the same base and between the same parallel lines then what is the area of parallelogram ABPD.
- The area of parallelogram PQRS is 88 cm sq. A perpendicular from S is drawn to intersect PQ at M. If SM = 8 cm, then find the length of PQ.
- Amy needs to order a shade for a triangular-shaped window that has a base of 6 feet and a height of 4 feet. What is the area of the shade?
- Monica has a triangular piece of fabric. The height of the triangle is 15 inches and the triangle’s base is 6 inches. Monica says that the area of the fabric is 90 square inches. What error did Monica make? Explain your answer.
- The sixth-grade art students are making a mosaic using tiles in the shape of right triangle. The two sides that meet to form a right angle are 3 centimeters and 5 centimeters long.
If there are 200 tiles in the mosaic, what is the area of the mosaic? - A parallelogram with area 301 has a base of 35. What is its height?
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