Chapter One Vocab and important concepts
Point – has no actual size, used to represent a location in space
Line – has no thickness or width, used to represent a continuous set of linear points that extend
indefinitely in both directions
Plane – has no thickness, used to represent a flat surface that extends indefinitely in all directions
Collinear – set of points that lie upon the same line
Non-collinear – set of points that do not lie on the same line
Coplanar – set or points or lines that lie in the same plane
Non-coplanar – set of points or lines that do not lie in the same plane
Skew lines – non-coplanar lines that do not intersect
Parallel lines – coplanar lines that do not intersect
Line segment – has two definite endpoints and can be measured, unlike a line
Betweenness of points – point M is between points P and Q if and only if, P,Q, and M are collinear and
PM+MQ=PQ
Congruent – line segments or angles that have the same measure
Midpoint – is the halfway point between the endpoints of a segment
Segment bisector – any segment, line, or plane that bisects a segment at its midpoint
Degree – unit of measure for dividing the circumference of a circle into 360 equal parts (1\360 of a turn
around a circle)
Ray – part of a line. It has one endpoint and extends indefinitely in one direction
Opposite rays – a point on a line determines exactly two rays called opposite rays. Opposite rays form a
straight angle.
Angle – is formed by two non-collinear rays that have a common endpoint. The rays are called the sides
of the angle. The common endpoint is called the vertex.
Right angle – an angle with a measure of 90 degrees
Acute angle – an angle with a measure that is less than 90 degrees
Obtuse angle – an angle with a measure greater than 90 degrees but less than 180 degrees
Straight angle – an angle with a measure of 180 degrees
Angle bisector – a ray (or line segment) that divides an angle into two congruent angles
Adjacent angles – are two angles that lie in the same plane, have a common vertex and a common side,
but no common interior points
Vertical angles – are two non adjacent angles formed by two intersecting lines
Linear pair of angles- is a pair of adjacent angles with non common sides that are opposite rays
Complementary angles – are two angles with measures that have a sum of 90 degrees
Supplementary angles – are two angles with measures that have a sum of 180 degrees
Perpendicular – lines, segments, or rays that form right angles
Polygon – is a closed figure whose sides are all segments
Regular polygon – a convex polygon in which all sides and all angles are congruent
Perimeter of a polygon – is the sum of the length of the sides of the polygon
Circumference – is the distance around a circle
Area – is the number of square units needed to cover a surface
Polyhedron – is a solid with all flat surfaces that enclose a single region of space. Each flat surface (or
face) is a polygon. The line segments where the faces intersect are called edges. Edges intersect at a
point called a vertex.
Prism – is a polyhedron with two parallel congruent faces called bases. The intersection of three edges
is called a vertex. Prisms are named by the shape of the bases.
Regular prism – is a prism with bases that are regular polygons.
Pyramid – a polyhedron with all faces (except one) intersecting at one vertex. Pyramids are named by
their bases which can be any polygon.
Regular polyhedron – if all of its faces are regular congruent polygons and all edges are congruent
There are only 5 regular polyhedrons and they are called Platonic solids. (see page 61)
Solids that are not polyhedrons:
Cylinder – is a solid with congruent circular bases in a pair of parallel planes
Cone – has a circular base and a vertex
Sphere – is a set of points in space that are given distance from a given point
Surface area – is the sum of the areas of each face of a solid
Volume – is the measure of the amount of space the solid encloses
Postulates/Key Concepts
Through any two points there is exactly one line
Through any three non-collinear points, there is exactly one plane
A line contains at least 2 points
A plane contains at least three points not on the same line
If two points lie in a plane, the line containing those two points also lies in the plane
If two lines intersect, they intersect in exactly one point
If two planes intersect, their intersection is a line
Distance on a number line for a segment with endpts A and B is given by: |A-B|
In the coordinate plane, the distance between two points can be found using the distance formula as follow:
x2 and y2 are the x,y coordinates for one point
x1and y1 are the x,y coordinates for the second point
d is the distance between the two points
Midpoints:
On a number line, the coordinate of the midpoint of a segment with endpoints A and B is: a + b
2
For the midpoint between two points in the coordinate system is found by:
The midpoint M of the line segment joining the
points (x1, y1) and (x 2, y 2) is
Find the midpoint of line segment
A(-3,4)
B(2,1)
The midpoint will have coordinates
Answer
.
To find and endpoint of a segment in the plane when given one endpoint and the
midpoint.
Consider this "tricky" midpoint problem:
M is the midpoint of
. The coordinates M(-1,1) and C(1,-3) are given.
Find the coordinates of point D.
First, visualize the situation. This will give you an idea of approximately where point D will be
located. When you find your answer, be sure it matches with your visualization of where the
point should be located.
Solve algebraically:
M(-1,1), C(1,-3) and D(x,y)
Substitute into the Midpoint Formula:
Solve for each variable separately:
The coordinates of
point D are (-3,5).