1.2 Key Concepts

PostulateA rule that is accepted without

proof. Sometimes called an Axiom.

Theorem A rule that can be proved.

Congruent Segments

Line Segments that have equal lengths

Ruler Postulate The points on a line can be matched one to one with real

numbers.

The number that corresponds to a point is the

COORDINATE of that point.

Ruler Postulate The DISTANCE between points is the absolute value

of the difference of the coordinates.Example

EXAMPLE 1 Apply the Ruler Postulate

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

SOLUTIONAlign one mark of a metric ruler with S. Then estimate the coordinate of T. For example, if you align S with 2, T appears to align with 5.4.

Use Ruler Postulate.ST = 5.4 – 2 = 3.4

The length of ST is about 3.4 centimeters.ANSWER

EXAMPLE 2 Apply the the Segment Addition Postulate

SOLUTION

MapsThe cities shown on the map lie approximately in a straight line. Use the given distances to find the distance from Lubbock, Texas, to St. Louis, Missouri.

Because Tulsa, Oklahoma, lies between Lubbock and St. Louis, you can apply the Segment Addition Postulate.

LS = LT + TS = 380 + 360 = 740

The distance from Lubbock to St. Louis is about 740 miles.

ANSWER

GUIDED PRACTICE for Examples 1 and 2

Use a ruler to measure the length of the segment to the nearest inch.1

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1.

2.

GUIDED PRACTICE for Examples 1 and 2

3. Use the Segment Addition Postulate to find XZ.

In Exercises 3 and 4, use the diagram shown.

ANSWER 73

In the diagram, WY = 30. Can you use the Segment Addition Postulate to find the distance between points W and Z? Explain your reasoning.

4.

No; W is not between X and Z.ANSWER

EXAMPLE 3 Find a length

Use the diagram to find GH.

Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH.

SOLUTION

Segment Addition Postulate.

Substitute 36 for FH and 21 for FG.

Subtract 21 from each side.

21 + GH=36

FG + GH=FH

=15 GH

EXAMPLE 4 Compare segments for congruence

SOLUTIONTo find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints.

Use Ruler Postulate.JK = 2 – (– 3) = 5

Plot J(– 3, 4), K(2, 4), L(1, 3), and M(1, – 2) in a coordinate plane. Then determine whether JK and LM are congruent.

EXAMPLE 4 Compare segments for congruence

To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints.

Use Ruler Postulate.LM = – 2 – 3 = 5

JK and LM have the same length. So, JK LM.=~

ANSWER

GUIDED PRACTICE for Examples 3 and 4

5. Use the diagram to find WX.

107ANSWER

No.ANSWER

Plot the points A(– 2, 4), B(3, 4), C(0, 2), and D(0, – 2) in a coordinate plane. Then determine whether AB and CD are congruent.

6.