Angles and SegmentsSections 1.4-1.6

Students will be able to…• Identify the bisector of an angle and determine if 2 adjacent angles are congruent• Define and identify a perpendicular bisector• Identify the midpoint of a segment• Determine if 2 segments are congruent• Find the length of segments using the distance formula

2 segments with the same length are:CONGRUENT SEGMENTS ( )

AB = CD DON’T SAY Equal signs only compare numbers, never geometric figures

A B DC

2 cm 2 cm

CDABCDAB

Find the length of each segment.

WHICH TWO SEGMENTS ARE ?

A B EDC

AB=

BC=

CE=

CD=

CDAB

Midpoint: divides a segment into 2 congruent segments

A midpoint, or any line, ray, or other segment through a midpoint, is said to BISECT the segment. (divides into 2 = parts)

B is the midpoint;

A B C

BCAB

You can use the definition of a midpoint to find lengths.

C is the midpoint of AC = 2x +1CB = 3x -4

Since we know by definition that AC = CB, set the expressions = to each other and solve for x.

AB

Find x, RM and MT.

M TR

5x+9 8x-36

Segment Addition Postulate

A B C

AB + BC = ACAB = 2BC = 10AC = 2 + 10 = 12

THE SUM OF THE PARTS EQUAL THE WHOLE!!

ANGLE ADDITION POSTULATE

ABCmDBCmABDm

THE SUM OF THE PARTS EQUAL THE WHOLE!!

ANGLES THAT FORM A STRAIGHT LINE ADD UP TO 180°

Example. Solve for the variable.

Perpendicular Lines

• Two lines that intersect to form right angles• The symbol means “is perpendicular to”

CDAB

Perpendicular Bisector

• Segment, line or ray to the segment at its midpoint

• It bisects the segment into 2 congruent segments

Congruent Angles Angles can be marked to show they are congruent

using arcs at the vertex Congruent angles will have the same number of arcs

Angle Bisector• A ray that divides an angle into two congruent, coplanar

angles• Its endpoint is at the angle’s vertex

L

NK J

bisects Therefore, KN

JKNLKN

EXAMPLE:

bisects

Find theHint: Draw the angle first. Then label given information.

WR AWB

AWBm

484 xBWRmxAWRm

The Distance FormulaUsed to find the distance between 2

points (or the length of segment between the 2 points): A( x1, y1) and B(x2, y2)

You also could just plot the points and use the Pythagorean Theorem!!

2122

12 )( yyxxd

Find the distance between the two points. Round your answer to the nearest tenth.

1. T(5, 2) and R(-4, -1)

Graph the 2 points on the coordinate plane. Then find the length of segment AB.

A( -2, -3) and B(1, 3)

Why are these pairs of points different??1. (2, 5) and (2, 9)

2. (-4, 7) and ( 3, 7)

Midpoint FormulaFind the midpoint coordinates

between 2 pointsFind by averaging the x-

coordinates and the y-coordinates of the endpoints

2,

22121 yyxxM

(x1, y1)

(x2, y2)

Use the following segment to answer the questions:

1. What is the length of the segment?

2. What are the coordinates of the midpoint?

Is R the midpoint of QT? Justify your answer.