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PSAT Prep: Tutoring Solution17 chapters | 173 lessons

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Lesson Transcript

Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

In this lesson, we'll discuss the meaning of postulates in mathematics and explore some common examples. We'll also apply these postulates to solving some geometric problems.

A **postulate** is a statement that is accepted without proof. **Axiom** is another name for a postulate. For example, if you know that Pam is five feet tall and all her siblings are taller than her, you would believe her if she said that all of her siblings are at least five foot one. Pam just stated a postulate, and you just accepted it without grabbing a tape measure to verify the height of her siblings.

**Conjectures** are often confused with postulates. Conjectures are conclusions that we make based on things that we observe. For example, if from Sunday to Thursday, Sam usually had pancakes for breakfast, you'd be safe in assuming he'd have pancakes on Friday and Saturday. However, on Friday, Sam may just have oatmeal. While conjectures may need to be proven before they're accepted, postulates are givens and need no proof. Every mathematical **theorem** began as a conjecture or a postulate before they were tested and accepted as proven mathematical facts, such as the ones we'll explore below.

The following are some postulates that apply to the four operations, including addition, subtraction, multiplication, and division. These postulates are also algebraic properties used to solve algebraic equations.

**The Addition Postulate**: If you have one apple and Sally has one apple, when you both add the same quantity to your existing number of apples, you'll still have the same number of apples. Using algebra, the postulate states:

If *x* = *y*, then *x* + 4 = *y* + 4

**The Subtraction Postulate**: If you have ten apples and Sally has ten apples, when you both subtract the same quantity of apples from your existing number of apples, you'll still have the same number of apples.

If *x* = *y*, then *x* - 3 = *y* - 3

Without being repetitive, these same principles apply to both multiplication and division.

**The Multiplication Postulate**: If *x* = *y*, then *x* * 3 = *y* * 3

**The Division Postulate**: If *x* = *y*, then *x* / 7 = *y* / 7

Geometric postulates can help us solve problems with lines, line segments, and angles. Let's see what they say.

**The Ruler Postulate**: Points on a line can match up with real numbers. In other words, each point on the line will represent a real number.

**The Segment Addition Postulate**: Remember that a segment has two endpoints. If you have a line segment with endpoints *A* and *B*, and point *C* is between points *A* and *B*, then *AC* + *CB* = *AB*.

**The Angle Addition Postulate**: This postulates states that if you divide one angle into two smaller angles, then the sum of those two angles must be equal to the measure of the original angle. Formally, if ray *QS* divides angle *PQR*, then the measure of the angles *PQS*, plus the measure of angle *SQR*, is equal to the measure of angle *PQR*.

The segment addition postulate and the angle addition postulate are called **partition postulates**. The idea behind partition postulates is this: if you cut your slice of bread into four pieces, when added together, those four pieces must form the whole slice. Similarly, if you divide a line or angle into two parts, then the sum of those two parts must be equal to the whole line or the whole angle.

Postulates are used to complete geometric proofs and solve mathematical problems. Let's explore the segment addition postulate and take a look at two common types of math problems using the angle and segment addition postulates.

Line *AB* is 42 centimeters (cm) long. Point *C* is on line *AB*, and line segment *AC* is 18cm long. So, how long is *CB*?

To make the problem more interesting, let's assume that point *C* is not the midpoint of line *AB*, which means it is not in the middle of the line. In using the segment addition postulate, we know that if *C* is between the endpoints *A* and *B*, then *AC* + *CB* = *AB*. All we have to do now is substitute the letters in the equation with numbers, so 18 + *CB* = 42. We then apply the subtraction postulate by subtracting 18 from both sides of the equation, so *CB* = 24 cm.

Let's solve another problem involving angles instead of lines.

According to the illustration, we see that angle *EFG* = 125. If the smaller angle *EFH* = 57, then using the subtraction postulate, we see that the other portion, angle *HFG*, is 68 degrees.

Let's review. A **postulate** is a statement accepted to be true without proof. Some common algebraic properties are also postulates and deal with the four operations: addition, subtraction, multiplication, and division. Other true statements are geometric postulates and include the **ruler**, **angle addition**, and **segment addition** postulates. Postulates are used particularly in proofs and sometimes in math problems. They form the basis for constructing many mathematical ideas and theorems.

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PSAT Prep: Tutoring Solution17 chapters | 173 lessons

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