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
![Segments and Congruence](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2189.png)
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.
![cordinate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2191-1024x126.png)
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
![Postulate 1 Ruler Postulate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2190.png)
![Postulate 1 Ruler Postulate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2215.png)
![Postulate 1 Ruler Postulate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2192.png)
Apply the Ruler Postulate
The below diagram shows the application of ruler postulate:
![Apply the Ruler Postulate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2193.png)
Example 1:
Measure the length of CD to the nearest tenth of a centimeter.
![Example1:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2194.png)
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
![Example 1: solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2216.png)
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.
![Postulate 2 Segment Addition Postulate:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2195.png)
Apply the Segment Addition Postulate
Segment addition postulate is adding two pieces of a segment and total the whole.
![Apply the Segment Addition Postulate](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2196.png)
Example 2:
Road Trip: The locations shown lie in a straight line.
Find the distance from the starting point to the destination.
![Example 2:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2197.png)
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
.
![Find a length Example 3:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2198.png)
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.
![Compare segments for congruence](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2199-1024x271.png)
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.
![Compare segments for congruence example 4 solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2200.png)
Exercise
- Look at the image below
![Look at the image below](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2217.png)
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.
![Use the number line given below and find the indicated distance.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2218.png)
- JK
- JL
- JM
- KM
- Find the lengths of AB to the nearest 1/8 inch.
![Find the lengths of AB to the nearest 1/8 inch.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2219.png)
- Find QS and PQ.
![Find QS and PQ.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2220.png)
- Use the diagram to find.
![Use the diagram to find.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2221.png)
- In the below figure, AC = 14 and AB = 9.
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2223.png)
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.
![L In the diagram, points V, W, X, Y, and Z are collinear, VZ = 52, XZ = 20, and WX =
XY =YZ.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2224.png)
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
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-2204-1024x406.png)
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