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Wize High School Geometry Textbook (Common Core) > Reasoning and Proofs

Postulates

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Postulates


In geometry there are a few statements we can take to be true. Let's learn about them

Postulates

A postulate is a statement that is assumed to be true. You can use these as the foundation or starting point for complex statements and arguments.

Postulates about Measuring and Adding

  • Line Measuring Postulate - Points on a line can be paired up with the real line. In this way we can measure points on lines and segments.
  • Angle Measuring Postulate - The rays of an angle can be matched up with a protractor. In this way we can measure angles.
  • Adding Segments Postulate - If points A,B,A, B, and CC are all on a line, with BB between AA and CC, then AB‾+BC‾=AC‾\overline{AB} + \overline{BC} = \overline{AC}. If we are given that AB‾+BC‾=AC‾\overline{AB} + \overline{BC} = \overline{AC} then it must be that point BB is between AA and CC.
  • Adding Angles Postulate - If point DD is in the interior of angle ∠ABC\angle ABC, then m∠ABD+m∠DBC=m∠ABCm\angle ABD + m\angle DBC = m\angle ABC

Postulates about points, lines, and planes.

Here are a few more postulates that we have seen so far:
  • Through any two points, there is exactly one line.
  • A line contains at least two points.
  • If two lines intersect, the intersection is a point.
  • Through any three points that are not on the same line, there is exactly one plane.
  • A plane contains at least three points, not on the same line.
  • If there are two points in a plane, then the line containing them is also in the plane.
  • If two planes intersect, the intersection is a line.
Example
For each of the postulates above, draw an example.
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Reasoning with Postulates

At your school, there are a total of 5 different sidewalks, as seen in the diagram below.


1. What is the smallest number of students needed such that every sidewalk has at least two students on it?

ANSWER: If placed carefully at the intersection of sidewalks, it will take 5 students to have at least two students on each sidewalk. One possible arrangement would be:


2. What postulate is being modeled in this situation?

ANSWER: This models the postulate that a line contains at least two points. The sidewalks are the lines, and the students are the points. Note that a line may contain more than two points, but the key phrase used says s "at least two points."

Postulates visualized


Having a diagram or picture for each of the postulates is a great way to get familiar with them.
Match each diagram with the corresponding postulate.
A.
If points A,B,A, B, and CC are all on a line, with BB between AA and CC, then AB‾+BC‾=AC‾\overline{AB} + \overline{BC} = \overline{AC}.

B.
A line contains at least two points.

C.
Through any three points that are not on the same line, there is exactly one plane.

D.
If two lines intersect, the intersection is a point.

Interpreting a diagram


Often a diagram can give us a lot of information about how points, lines, and planes are related to one another.
We can use a diagram, along with postulates to make certain assumptions.

Use the diagram to mark each statement as "Assumed true" or "Can not assume."


1. Plane ABDABD and plane BDCBDC meet at line BD↔\overleftrightarrow{BD}.

2. Line CD↔\overleftrightarrow{CD} is in the plane BDCBDC.

3. Lines AB↔\overleftrightarrow{AB} and CD↔\overleftrightarrow{CD} intersect.
1. Plane ABDABD and plane BDCBDC meet at line BD↔\overleftrightarrow{BD}.

The hands of a clock


The hands of a clock form various angles throughout the day. Use the diagram of a clock below the answer the following questions.


1. Suppose that we are given that m∠ABD=64∘m\angle{ABD} = 64^\circ, and that m∠ABC=150∘m\angle{ABC} = 150^\circ.
What is the name of the postulate that can help us solve for the measure of angle ∠DBC\angle{DBC} ?

2. What is the measure of angle ∠DBC\angle{DBC}?
1. Suppose that we are given that m∠ABD=64∘m\angle{ABD} = 64^\circ, and that m∠ABC=150∘m\angle{ABC} = 150^\circ.
What is the name of the postulate that can help us solve for the measure of angle ∠DBC\angle{DBC} ?