1 Lesson 1-2 Use Segments and Congruence. Warm Up Problems 2 Solve for x.

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Lesson 1-2

Use Segments

and Congruence

Warm Up Problems

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Solve for x.

214102 xx

6084163 xx

xx 3101782

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Theorem

Definition: A true statement that follows as a result of other true statements. A statement that can be proven.

We will investigate theorems in greater detail later.

Your book lists all the Theorems (and Postulates ) we will use this year. Page 926

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Postulate or axiom

Definition: A rule that is accepted without proof.

Examples:

• Through any two points there is exactly one line.

• Through any three points, there is exactly one plane.

• A line contains at least two points.

• A plane contains at least three points.

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Postulates

• If two planes intersect, then the intersecting is a line.

• If two points lie in a plane, then the line containing the two points lie in the same plane.

Examples:

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POSTULATE 1 The Ruler Postulate

The Ruler Postulate: Points on a line can be matched one to one with the real numbers. • The real number that corresponds to a point is the coordinate of the point.

• The distance between points A and B, written AB, is the absolute value of the difference of the coordinates of A and B.

•x1 –small numbers are called subscripts (read “x sub one”)

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Ruler Postulate : Example

-5 5

SRQPOLKJIHG M N

PK = | 3 - -2 | = 5 Remember : Distance is always positive

Find the distance between P and K.

Note: The coordinates are the numbers on the ruler or number line! The capital letters are the names of the points.

Therefore, the coordinates of points P and K are 3 and -2 respectively.

Substituting the coordinates in the formula │a – b │

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Between

Definition: B is between A and C if AB + BC = AC.

AB + BC = AC AX + XB > AB

A BX

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POSTULATE 2 The Segment Addition Postulate

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

Example: If AB = x , BC = 2x and AC = 12, then, find x, AB and BC.

AB + BC = AC

x + 2x = 12

3x = 12

x = 4

2xx

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x = 4AB = 4BC = 8

Step 1: Draw a figure

Step 2: Label fig. with given info.

Step 3: Write an equation

Step 4: Solve and find all the answers

AC

B

Apply the Segment Addition Postulate

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SOLUTION

Maps

The 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

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Congruent Segments

Definition:

If numbers are equal the objects are congruent.

AB: the segment AB ( an object )

AB: the distance from A to B ( a number )

AB

D

C

Congruent segments can be marked with dashes.

Correct notation:

Incorrect notation:

AB = CD AB CD

AB = CDAB CD

Segments with equal lengths. (congruent symbol: )

Compare segments for congruence

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SOLUTION

To 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.

Compare segments for congruence (continued)

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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

Homework #2

p. 878: 7-12

p. 12: 1-8, 12-23, 32-34, 37-45

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