Points, Lines, and PlanesSections 1.1 & 1.2

Definition: PointA point has no dimension.

It is represented by a dot.

A point is symbolized using an upper-case letter.

Definition: Line

A line has one dimension. (infinite length)

Name a line using any 2 points on the line with a two sided arrow above:

Also name by using a lower-case cursive letter.

Line l

Definition: PlaneA plane has 2 dimensions. It is represented by a shape that looks like a parallelogram. It extends infinitely in length and width.

Name a plane using the word plane with 3 non-collinear points in the plane.

Plane ABC

Also name with an upper-case cursive letter.Plane M

Definition: Collinear Points Points that lie (or could lie) on the same line.

Definition: CoplanarCoplanar points are points that lie (or could lie) in the same plane.

Definition: Line SegmentA line segment consists of two endpoints and all the points between them.

Named using both endpoints with a line segment above like this: .

and refer to the same line segment.

Definition: RayThe ray consists of an endpoint and all points on a line in the opposite direction.

A ray is named using its endpoint first and then any other point on the ray with a ray symbol pointing to the right above them like this: .

and do not refer to the same ray.

Definition: Opposite RaysIf point C lies on line AB between A and B, then ray CA and ray CB are opposite rays.

Two opposite rays make a line.

Definition: IntersectionThe intersection of two or more figures is the set of points the figures have in common.

The intersection of 2 different lines is a point.

The intersection of 2 different planes is a line.

Definition: PostulateA rule that is accepted without proof.

Definition: TheoremA rule that can be proven.

Definition: Between Between also implies collinear.

Definition: Congruent SegmentsLine segments of equal (=) length are called congruent (segments.

To show that two segments are congruent in a drawing we use matching tick marks.

A B C D

Definition: DistanceThe distance between points A and B is also known as the length of line segment AB. Distance is how many units apart the points lie. The distance from A to B, or the length of is symbolized as AB. (No symbol above).

Distance Formula The distance formula is used to compute

the distance between two points in a coordinate plane. It is given by:

2 22 1 2 1( ) ( )d x x y y

Finding the Distance Find the distance between the points (1,

4) and (-2, 8).

Alternative to the Distance Formula The distance formula comes from the

Pythagorean theorem: a2 + b2 = c2

If you are unsure about the distance formula, graph the two points accurately on a graph and use the Pythagorean theorem to find the distance.

Finding distance Find the distance between (-2, 3) & (10,

8) by graphing and using the Pythagorean theorem.

Compare the two ways Find the distance

between (-7, -3) & (8, 5) using the distance formula.

Graph the same two points and find the distance using the Pythagorean Theorem.

Segment Addition PostulateIf B is between A and C, then AB + BC = AC.

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

Definition: MidpointThe midpoint of a segment is the point that divides the segment into two congruent pieces.

Midpoint Formula The coordinates of the midpoint of a

segment are the averages of the x-coordinates and of the y-coordinates of the endpoints.

1 2 1 2,2 2

x x y yM

Finding a Midpoint Find the midpoint

between the endpoints (1, 7) & (3, -4).

Find the midpoint between the endpoints (2, 5) & (-3, 9)

Finding an Endpoint If the midpoint of

segment AB is (2, 3) and A is at (-1, 5), where is B located?

If the midpoint of segment CD is (0, -2) and D is at (3, 4), where is C located?

Definition: Segment Bisector A segment bisector is a point, ray,

line, line segment, or plane, that intersects the segment at its midpoint.

Definition: AngleFormed by two different rays with the same endpoint called the vertex.

An angle is named using 1) three points with the vertex in middle2) just the vertex iff no other angle has the same vertex3) a number assigned to the angle

Definition: Measure of an angleTo denote the measure of an angle, we write an “m” in front of the angle sign: o

Definitions: Angles Classified by MeasureAn acute angle has a measure between 0o and 90o

A right angle has a measure of exactly 90o

An obtuse angle has a measure between 90o and 180o

A straight angle has a measure of 180o

Angle Addition PostulateThe measures of two adjacent angles can be added to represent the large angle they form.

Definition: Angle BisectorAn angle bisector is a ray that divides one angle into two congruent angles.

Definition: Congruent AnglesTwo angle are congruent if they have the same measure.

To show that two angles in a diagram are congruent, we put a matching arc inside each angle.

Definition: Complementary AnglesTwo angles whose measures sum to 90°

Definition: Supplementary Angles Two angles whose measures sum

to180o.

Definition: Adjacent AnglesTwo angles that share a common vertex and side, but have no common interior points.

Definition: Linear PairTwo adjacent angles whose sides form a straight line.

The angles in a linear pair are always supplementary .

Definition: Vertical Angle PairsFormed when two lines intersect. The angle pairs only touch at the vertex.

There are two pairs of vertical angles formed whenever two lines intersect.