## Key Concepts

- Apply the ruler postulate
- Apply the segment addition postulate
- Find a length
- Compare segments for congruence

### Introduction

In this chapter, we will learn to apply the ruler postulate, apply the segment addition postulate, find a length and compare segments for congruence.

What are postulates/axiom?

A rule in geometry that is accepted without proof, it is given.

What is a theorem?

A rule that can be proved.

**Example:** Pythagorean Theorem

**What is coordinate? **

The real number that corresponds to a point. It is a set of values that shows an exact position on the scale or graph.

**What is distance? **

**Positive length between two points. **

**What is ruler postulate? **

**The distance between two points is the absolute value of the difference between them. **

### Postulate 1 Ruler Postulate

#### Apply the Ruler Postulate

The below diagram shows the application of ruler postulate:

**Example 1:**

Measure the length of CD to the nearest tenth of a centimeter.

**Solution:**

Align one mark of a metric ruler with C. Then estimate the coordinate of D. For example, if you align C with 1, D appears to align with 4.7

CD = | 4.7 – 1 | = 3.7 Ruler postulate

The length of CD is about 3.7 centimeters.

### Postulate 2 Segment Addition Postulate:

If B is between A and C, then

AB + BC = AC

If AB + BC = AC, then B is between A and C.

#### Apply the Segment Addition Postulate

Segment addition postulate is adding two pieces of a segment and total the whole.

**Example 2:**

Road Trip: The locations shown lie in a straight line.

Find the distance from the starting point to the destination.

**Solution:**

The rest area lies between the starting point and the destination, so you can apply the segment addition postulate.

SD = SR + RD à Segment addition postulate

= 64 + 87 à Substitute for SR and RD.

= 151 à Add.

The distance from the starting point to the destination is 151 miles.

#### Find a length

**Example 3:**

Use the diagram to find KL

.

**Solution:**

Use the segment addition postulate to write an equation.

Then solve the equation to find KL.

JL = JK + KL à Segment addition postulate

38 = 15 + KL à Substitute for JL and JK

23 = KL à Subtract 15 from each side.

#### Compare segments for congruence

What are congruent segments?

Line segments that have the same length.

**Example 4:**

Plot F(4, 5), G( -1, 5), H(3, 3), and J(3, -2) in a coordinate plane. Then determine whether FG and HJ are congruent.

**Solution:**

Horizontal segment:

Subtract the x-coordinates of the endpoints.

FG = | 4 – (-1) | = 5

Vertical segment:

Subtract the y-coordinates of the endpoints.

HJ = | 3 – (-2) | = 5

FG and HJ have the same length. So FG = HJ.

## Exercise

- Look at the image below

Explain how you can find PN if you know PQ and QN.

How can you find PN if you know MP and MN?

- Use the number line given below and find the indicated distance.

- JK
- JL
- JM
- KM

- Find the lengths of AB to the nearest 1/8 inch.

- Find QS and PQ.

- Use the diagram to find.

- In the below figure, AC = 14 and AB = 9.

Describe and correct the error made in finding BC.

- L In the diagram, points V, W, X, Y, and Z are collinear, VZ = 52, XZ = 20, and WX =

XY =YZ.

Find the indicated length

- Find the length of WX
- Find the length of VW
- Find the length of WY
- Find the length of VX
- Find the length of WZ
- Find the length of VY

- Consider the points A(-2, -1), B(4, -1), C(3, 0), and D(3, 5). Are and congruent?
- Plot J(-3, 4), K(2, 4), L(1, 3), and M(1, -2) in a coordinate plane. Then determine whether segment and segment are congruent.
- Plot the points
*A*(0, 1),*B*(4, 1),*C*(1, 2),*D*(1, 6) in a coordinate plane. Determine whether the line segments and are congruent.

### What have we learned

- Apply the ruler postulate to a line and measure its length.
- Apply the segment addition postulate and find the total length
- Compare two or more segments for congruence.

### Concept Map

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