# Notes

advertisement ```Points, Lines,
and Planes
Sections 1.1 &amp; 1.2
Definition: Point
A 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:
𝐴𝐵
Also name by using a lower-case cursive letter.
Line l
Definition: Plane
A 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 noncollinear 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: Coplanar
Coplanar points are points that lie (or could
lie) in the same plane.
Definition: Line Segment
A line segment consists of two endpoints
and all the points between them.
Named using both endpoints with a line
segment above like this: 𝐴𝐵.
𝐴𝐵 and 𝐵𝐴 refer to the same line segment.
Definition: Ray
The 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: 𝐴𝐵.
𝐴𝐵 and 𝐵𝐴 do not refer to the same ray.
Definition: Opposite Rays
If 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: Intersection
The 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: Postulate
A rule that is accepted without proof.
Definition: Theorem
A rule that can be proven.
Definition: Between
Between also implies collinear.
Definition: Congruent Segments
Line 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: Distance
The 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
𝐴𝐵 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:
d  ( x2  x1 )  ( y2  y1 )
2
2
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) &amp; (10, 8)
by graphing and using the Pythagorean
theorem.
Compare the two ways
 Find
the distance
between (-7, -3) &amp;
(8, 5) using the
distance formula.
 Graph
the same
two points and find
the distance using
the Pythagorean
Theorem.
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
Definition: Midpoint
The 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 xcoordinates and of the y-coordinates of
the endpoints.
 x1  x2 y1  y2 
M 
,

2 
 2
Finding a Midpoint
 Find
the midpoint
between the
endpoints (1, 7) &amp;
(3, -4).
 Find
the midpoint
between the
endpoints (2, 5) &amp;
(-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: Angle
Formed by two different rays with the same
endpoint called the vertex.
An angle is named using ∠ and
1) three points with the vertex in middle
2) just the vertex iff no other angle has the
same vertex
3) a number assigned to the angle
Definition: Measure of an
angle
To denote the measure of an angle, we
write an “m” in front of the angle sign:
m∠𝐴𝐵𝐶 = 90o
Definitions: Angles Classified
by Measure
An 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 Postulate
The measures of two adjacent angles can
be added to represent the large angle they
form.
Definition: Angle Bisector
An angle bisector is a ray that divides one
angle into two congruent angles.
Definition: Congruent Angles
Two 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
Angles
Two angles whose measures sum to 90&deg;
Definition: Supplementary
Angles
 Two
angles whose measures sum to180o.
Definition: Adjacent Angles
Two angles that share a common vertex
and side, but have no common interior
points.
Definition: Linear Pair
Two adjacent angles whose sides form a
straight line.
The angles in a linear pair are always
supplementary .
Definition: Vertical Angle Pairs
Formed when two lines intersect. The angle
pairs only touch at the vertex.
There are two pairs of vertical angles formed
whenever two lines intersect.
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