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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 cm
^{2}, 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 cm^{2}, then find the area of parallelogram PQRS. - ABCD is a parallelogram where E is a point on AD. Area of ΔBCE = 21 cm
^{2}. 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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