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1Lesson 4: Measuring Segments and Angles

Graph and label the following points on a coordinate grid.P(-1, -1), Q(0, 4), R(-3, 5), S(2, 5), and T(3, -4)

1.Name three noncollinear points.

2.Name three collinear points.

3.Name two intersecting lines

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Lesson 1-4

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Measuring Segments and Angles

Lesson 4: Measuring Segments and Angles

The Ruler Postulate (1-5)

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The Ruler Postulate: Points on a line can be paired with the real numbers in such a way that:

• Any two chosen points can be paired with 0 and 1.

• The distance between any two points on a number line is the absolute value of the difference of the real numbers corresponding to the points.

Formula: Take the absolute value of the difference of the two coordinates a and b: │a – b │


Ruler Postulate : Example

-5 5

SRQPOLKJIHG M N

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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 │


Between

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Definition: X is between A and B if AX + XB = AB.

A BX

AX + XB = AB AX + XB > AB

A BX


The Segment Addition Postulate

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AB

C

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

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

AC + CB = AB

x + 2x = 12

3x = 12

x = 4

2xx

12

x = 4AC = 4CB = 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


Congruent Segments

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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: )


Midpoint

a b

2

1 1( , )x y

2 2( , )x y

1 2 1 2,2 2

x x y y

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A point that divides a segment into two congruent segments

Definition:

EDFIf DE EF , then E is the midpoint of DF.

On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is .

In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and

is .

Formulas:


Midpoint on Number Line - Example

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Find the coordinate of the midpoint of the segment PK.

-5 5

SRQPOLKJIHG M N

a b 3 ( 2) 10.5

2 2 2

Now find the midpoint on the number line.


Segment Bisector

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Any segment, line or plane that divides a segment into two congruent parts is called segment bisector.

Definition:

B

E

D

FA

BE

D

FA

E

D

A F

B

AB bisects DF. AB bisects DF.

AB bisects DF.Plane M bisects DF.


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Angle and Points

vertex

ray

ray Angles can have points in the interior, in the exterior or on the angle.

Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.

A

BC

DE


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Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide

ABC or CBA

Using 3 points: vertex must be the middle letter

This angle can be named as

Using 1 point: using only vertex letter

* Use this method is permitted when the vertex point is the vertex of one and only one angle.

Since B is the vertex of only this angle, this can also be called .

A

B C

B


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Naming an Angle - continued

Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named

as .2

* The “1 letter” name is unacceptable when …more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.

2

A

B C


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Example

KTherefore, there is NO in this diagram.There is , ,LKM PKM and LKP

2 3 5!!!There is also and but there is no

K is the vertex of more than one angle.


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4 Types of Angles

Acute Angle: an angle whose measure is less than 90.

Right Angle: an angle whose measure is exactly 90 .

Obtuse Angle: an angle whose measure is between 90 and 180.

Straight Angle: an angle that is exactly 180 .


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Measuring Angles

Just as we can measure segments, we can also measure angles.

We use units called degrees to measure angles.

• A circle measures _____

• A (semi) half-circle measures _____

• A quarter-circle measures _____

• One degree is the angle measure of 1/360th of a circle.

?

?

?

360º

180º

90º


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Adding Angles

22°

36°

21

D

B

C

A

Therefore, mADC = 58.

m1 + m2 = mADC also.


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Angle Addition Postulate

R

M K

W

The sum of the two smaller angles will always equal the measure of the larger angle.

MRK KRW MRW

Postulate:


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Example: Angle Addition

R

M K

W

3x + x + 6 = 90 4x + 6 = 90 – 6 = –64x = 84x = 21

K is interior to MRW, m MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.

3xx+6 Are we done?

mMRK = 3x = 3•21 = 63º

First, draw it!


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Angle Bisector

An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.

UK

j41°

41°

64

U

K53

Example: Since 4 6, is an angle bisector.


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3 5.

Congruent Angles

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Definition: If two angles have the same measure, then they are congruent.

Congruent angles are marked with the same number of “arcs”.

The symbol for congruence is

Example:


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Example

Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87,

then find: mPMY = _______ mLMY = _______

ML


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