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# Segments and Congruence

## 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.

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

1. 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?

1. Use the number line given below and find the indicated distance.
1. JK
2. JL
3. JM
4. KM
1. Find the lengths of AB to the nearest 1/8  inch.
1. Find QS  and PQ.
1. Use the diagram to find.
1. In the below figure, AC = 14 and AB = 9.

Describe and correct the error made in finding BC.

1. 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

1. Find the length of WX
2. Find the length of VW
3. Find the length of WY
4. Find the length of VX
5. Find the length of WZ
6. Find the length of VY
1. Consider the points A(-2, -1), B(4, -1), C(3, 0), and D(3, 5). Are  and congruent?
2. 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.
3. 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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