lesson 1.3: segments, rays, and distance

Lesson 1.3: Segments, Rays, and Distance

Pre-AP Geometry

Points, Lines, and Planes

Line Segment Two points (called the endpoints) and all the

points between them that are collinear with those two points

Named line segment AB, AB, or BA

line AB segment AB 

A B A B

Length of a segment

Length of BC is stated as BC. It is the distance between points B and C.

On a number line, length of a segment is found by subtracting the coordinates of the endpoints.

On a coordinate plane, length of a segment is found using the distance formula D = 2 2

2 1 2 1( ) ( )x x y y

Examples

Find the length between 5 and -3 on the number line

Find the distance of segment AB if A(-3, 5) and B(2, -7)

Postulates

Postulate: statement that is accepted without proof

Segment Addition Postulate If B is between A and C, then AB + BC = AC

Ruler Postulate1. The points on a line can be paired with the

real numbers in such a way that any two points can have the coordinates 0 and 1

2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

Examples

EG = 7x + 3 EF = 3x + 8 FG = 2x + 1

1. Find x:2. Find EG:3. Find EF:4. Find FG:

EF G

Segment Length terms

Congruent- two objects that have the same size and shape. We use the symbol to show that two objects ≅are congruent.

Congruent segments- two segments with equal lengths.Example: DE FG≅

Midpoint of a segment: a point that divides a segment into two congruent segments.

Midpoint formula: M = ( )

Segment bisector: A line, segment, ray, or plane which intersects a segment at its midpoint.

2 1 2 1,2 2

x x y y

Points, Lines, and Planes

Ray Part of a line that starts at a point and extends

infinitely in one direction. Initial Point

Starting point for a ray.

Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D “It begins at C and goes through D and on

forever”

Points, Lines, and Planes

Opposite Rays If C is between A and B, then CA and CB

are opposite rays. Together they make a line.

A BC

Lesson 1.4: Angles

Parts of an angle

Sides of an angle are made up of rays

The rays meet at a point called the vertex

vertexsides

Naming an angle

An angle can be named by the vertex, by the 3 points on the angle: the side, the vertex and the other side, or a number inside the angle.

The angle can be named ∠GHI, ∠IHG, ∠H, or ∠1

G

I

H

1

Classifying angles

Acute angle: Angle measuring greater than 0° and less than 90°.

Obtuse angle: Angle measuring greater than 90° and less than 180°

Right angle: An angle measuring exactly 90°

Straight angle: An angle measuring exactly 180°

Angle Postulates

Protractor Postulate:On AB in a given plane, choose any point O between A

and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with real numbers from 0 to 180 in a way such that:a. OA is paired with 0, and OB with 180b. If OP is paired with x, and OQ with y, the m∠POQ = │x - y │

Angle addition postulate:-If B lies on the interior of ∠AOC, then m ∠AOB +

m∠BOC = m∠AOC-If ∠AOC is a straight angle, then m∠AOB+m ∠BOC =

180.

Angle Vocabulary

Congruent Angles Two angles with equal measures

Adjacent angles Angles which share a vertex and a

common side, but no common interior points

Angle bisector A ray which divides an angle into two

congruent, adjacent angles

Congruence symbols and drawing conclusions

Do not assume anything in geometry. Just because two segments look equal does not mean that they are.

Postulates and Theorems Relating Points, Lines, and PlanesLesson 1.5

Pre-AP Geometry

Postulates

A point is defined by its location.

A line contains at least two points.

A plane contains at least three points not all in one line.

Space contains at least four points not all in one plane.

Postulates

Through any two points there is exactly one line.

Through any three points there is at least one plane and through any three non-collinear points there is exactly one plane.

If two points are in a plane, then the line that contains the point is in that plane.

If two planes intersect, then their intersection is a line.

Theorem

If two lines intersect, then they intersect in exactly one point.

Theorem

Through a line and a point not in the line there is exactly one plane.

Theorem

If two lines intersect, then exactly one plane contains the lines.

Review Quiz – True or False

1. A given triangle can lie in more than one

plane.

2. Any two points are collinear.

3. Two planes can intersect in only one point.

4. Two lines can intersect in two points.

Review Quiz – True or False

1. A given triangle can lie in more than one plane. False. Through a line and a point not in the line there is exactly one plane.

2. Any two points are collinear. True. 3. Two planes can intersect in only one

point. False. If two planes intersect, then they intersection is a line.

4. Two lines can intersect in two points. False. If two lines intersect, then they intersect in exactly one point.

Problem Set 1.5Written Exercises

p.25: # 1 – 13 (odd), 14 - 20