Ch 1.
Points, Lines, Planes, & Angles
I. Sec 1.1
Points, Lines, & Planes
Know how to:
• Identify and model points, lines, and planes.
• Identify collinear and coplanar points and intersecting lines and planes in space.
Vocabulary
Point
Line
Collinear
Plane
Coplanar
Undefined Terms
Space
Locus
A.
Point
A point 0-dimensional mathematical object that has neither shape or size.
Points are identified by a capital letter.
Points can be specified in n-dimensional space using n coordinates.
The basic geometric structures of higher dimensional geometry--the line,
plane, space, and hyperspace--are all built up of infinite numbers of points
arranged in particular ways.
1.
Line
A line is a straight one-dimensional figure having no thickness and
extending infinitely in both directions.
While lines are intrinsically one-dimensional objects, they may be
embedded in higher dimensional spaces. ex. 2 dimensions (x, y), or 3
dimensions (x, y, z), or n dimensions (n1, n2, ..., nn)
A line is uniquely determined by two points, and the line passing through
points A and B points is denote AB. A line may also be denoted with a
single lower-case letter
AB
a.
B
A
Collinear Points
Three or more points A, B, C, ..., are said to be collinear if they lie on
a single straight line L. A line on which points lie.
Two points are trivially collinear since two points determine a line.
2.
Plane
A plane is a flat two-dimensional surface made up points.
A plane has no depth and extends infinitely in all directions.
Planes are identified by a capital script letter or by letters identifying 3
non-collinear points.
Z
B
C
A
a.
Coplanar Points
Points lying in a common plane are said to be coplanar.
Three noncollinear points determine a plane and so are trivially
coplanar.
Z
B
A
3.
C
Space
A boundless, three dimensional set of all points.
Space can contain lines and planes.
II. Sec 1.2
Linear Measure & Precision
Know how to:
• Measure segments and determine accuracy of measurement.
• Compute with measures
Vocabulary
Line Segment
Precision
Betweenness of Points
Between
Congruent
Construction
A.
Line Segment
A closed interval corresponding to a finite portion of an infinite line. The
interval is bounded by two collinear points.
Line segments are generally labeled with two letters corresponding to their
endpoints, say A and B, and then written AB or BA
The length of the line segment is indicated with an overbar.
A
B
AB
1.
Precision of Measurement
Precision of measurement is based on the smallest unit available on the
measurement tool.
Measurements should be precise to plus or minus one half of the smallest
unit available.
example; given a measurement of 19.4 cm made with a meter stick
indicates that the meter stick is graduated in centimeters and
millimeters. It also indicates that the actual measurement is between
19.35 cm and 19.45 cm.
2.
Calculated Measurement
For any three collinear points, one of the points is between the other
two outside points, and forms two adjacent line segments.
The sum of the two adjacent line segments is equal to the length of the
line segment defined by the two outside points.
A
B
C
AB + BC = AC
a.
Algebraic Example
A
B
C
AB + BC = AC
AB = 4w
find AB & AC
BC = 20
AC = 6w + 2
3.
Congruent Line Segments
Two line segments that have the same measure (length) are congruent.
J
I
H
K
m HI = 3.00 cm
m JK = 3.00 cm
HI  JK
HI is congruent to JK
III. Sec 1.3
Distance and Midpoints
Know how to:
• Find the distance between two points.
• Find the midpoint of a segment.
Vocabulary
midpoint
segment bisector
A.
Distance
Distance is the linear measure between two points.
1.
Number Line Distance
Given points R and S on a number line, where R = a and S = b.
The distance between R and S is |a - b| or |b - a|
2.
Coordinate Plane Distance
a.
Distance Formula
Given points A and B on a coordinate plane, where A = (x1, y1) and B
= (x2, y2)
The distance d between A and B is
d
3.
x2  x1  y2  y1
2
2
Synthetic Plane Distance
a.
Pythagorean Theorem
For any right triangle, where a and b are legs and c is the
hypotenuse,
B.
a 2  b 2  c2
Midpoints
A midpoint is a point that is exactly half way between two other points.
1.
Number Line Midpoint
Given points R and S on a number line, where R = a and S = b.
The coordinate m of the midpoint between R and S is
m
2.
ab
2
Coordinate Plane Midpoint
Given points A and B on a coordinate plane, where A = (x1, y1) and B =
(x2, y2)
The coordinate of the midpoint (xm,ym) between A and B is
1  x2 , y1  y2
x
xm,ym  
a.
2
2



Segment Bisector
Any segment, line, or plane that intersects a segment at its midpoint
is called a segment bisector.
3.
Synthetic Geometry Construction
Demonstrate construction of a segment bisector.
IV. Sec 1.4
Angle Measure
Know how to:
• Measure and classify angles.
• Identify and use congruent angles and the bisector of an angle.
Vocabulary
degree
ray
opposite rays
angle
sides
vertex
interior
exterior
right angle
acute angle
obtuse angle
angle bisector
A.
Degree
(unit of measure)
A degree is 1/360 of a turn around a circle.
It is one of the standard units for angular measure.
Angular measure can be determined by using a protractor.
B.
Ray
A ray is a part of a line that has one endpoint and extends infinitely in one
direction.
1.
Opposite Rays
Any point on a line creates two rays that extends infinitely in opposite
directions.
C.
Angle
An angle is formed by two noncollinear rays that have a common endpoint.
The rays are called the sides of the angle.
The common endpoint is called the vertex.
The angle interior is the region between the sides of the angle.
The angle exterior is the region outside of the angle interior.
1.
Acute Angle
An angle with an angular measure greater than zero and less than 90
degrees.
B
mBAC = 51.57 °
A
C
2.
Obtuse Angle
An angle with an angular measure greater than 90 degrees and less than
180 degrees.
B
mBAC = 100.15 °
A
C
3.
Right Angle
An angle with an angular measure equal to 90 degrees.
B
mBAC = 90.00 °
A
4.
C
Straight Angle
An angle with an angular measure equal to 180 degrees.
mBAC = 180.00 °
B
5.
A
C
Zero Angle
An angle with an angular measure equal to 0 degrees.
mBAC = 0.00 °
A
6.
B
C
Congruent Angles
Two angles with the same angular measures are congruent angles.
B
mBAC = 34.26 °
A
C
mCAD = 34.26 °
D
V. Sec 1.5
Angle Relationships
Know how to:
• Identify and use special pairs of angles
• Identify perpendicular lines.
Vocabulary
adjacent angles
vertical angles
linear pairs
complementary angles
supplementary angles
perpendicular
A.
Angle Pairs
Angle classification based on position
1.
Adjacent Angles
Two angles that lie in the same plane, have a common vertex, and a common side, but no
common interior points.
B
mBAC = 34.26 °
A
C
mCAD = 94.47 °
D
2.
Vertical Angles
Two nonadjacent angles formed by two intersecting lines.
B
mBAD = 114.42 °
D
mBAC = 65.58 °
A
C
mDAE = 65.58 °
E
mCAE = 114.42 °
3.
Linear Pair
A pair of adjacent angles whose noncommon sides are opposite rays.
C
mCAD = 130.14 °
mBAC = 49.86 °
D
A
B
B.
Angle Relationships
Angle classification based on summed angular measure.
1.
Complementary Angles
Two angles whose angular measures have a sum of 90 degrees.
Note - the two angles may or may not share vertices or sides.
B
mBAC = 24.02 °
C
D
A
E
F
mEDF = 65.99 °
mBAC+mEDF = 90.00 °
2.
Supplementary Angles
Two angles whose angular measures have a sum of 180 degrees.
Note - the two angles may or may not share vertices or sides.
B
mBAC = 24.02 °
C
F
D
A
E
mEDF = 155.99 °
mBAC+mEDF = 180.00 °
C.
Right Angle
1.
Perpendicular Lines
Two lines that intersect to form four 90 degree angles.
C
mCAD = 90.00 °
D
mBAC = 90.00 °
A
mDAE = 90.00 °
B
mBAE = 90.00 °
E
VI. Sec 1.6
Polygons
Know how to:
• Identify and name polygons
• Find perimeters of polygons
Vocabulary
polygon
concave
convex
n-gon
regular polygon
perimeter
A.
Polygon
A closed figure formed by a finite number of coplanar segments such that
• the sides that have a common endpoint are not collinear, and
• each side intersects exactly two other sides, but only at their endpoints.
A polygon is named by the letters of its vertices, written in consecutive order.
1.
n-gon Classification
Sides
3 triangle
4 quadrilateral
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 nonagon
10 decagon
11 hendecagon
12 dodecagon
2. Shape
Classification
Concave - a polygon such that there is a straight line that cuts it in four or more points
Convex - a polygon such that no side extended cuts any other side or vertex; it can be cut by
a straight line in at most two points
Regular polygon - a polygon which has congruent sides and congruent internal angles.
Irregular polygon - a polygon which is not a regular polygon
3.
Perimeter
The sum of the polygon side measures.