Transcript Unit 1 - My Teacher Pages

Unit 1
Describe and Identify the three
undefined terms, Understand Segment
Relationships and Angle Relationships
Part 1
Definitions:
Points, Lines, Planes and
Segments
Undefined Terms
Points, Line and Plane are all
considered to be undefined terms.
– This is because they can only be explained
using examples and descriptions.
– They can however be used to define other
geometric terms and properties
A
<
B
>
Point
– A location, has no shape or size
– Label:
Line
– A line is made up of infinite points and has no thickness or width, it will
continue infinitely.There is exactly one line through two points.
– Label:
Line Segment
– Part of a line
– Label:
Ray
– A one sided line that starts at a specific point and will continue on
forever in one direction.
– Label:
F
<
E
A
B
>
Collinear
– Points that lie on the same line are said to be
collinear
– Example:
Non-collinear
– Points that are not on the same line are said to be
non-collinear (must be three points … why?)
– Example:
Plane
– A flat surface made up of points, it has no depth
and extends infinitely in all directions. There is
exactly one plane through any three non-collinear
points
Coplanar
– Points that lie on the same plane are said to be
coplanar
Non-Coplanar
– Points that do not lie on the same plane are said
to be non-coplanar
Intersect
The intersection of two things is the
place they overlap when they cross.
– When two lines intersect they create a
point.
– When two planes intersect they create a
line.
Space
Space is boundless, three-dimensional
set of all points. Space can contain lines
and planes.
Practice
Use the figure to give examples of the following:
1.
2.
3.
4.
Name two points.
Name two lines.
Name two segments.
Name two rays.
5.
6.
7.
8.
9.
Name a line that does not contain point T.
Name a ray with point R as the endpoint.
Name a segment with points T and Q as its endpoints.
Name three collinear points.
Name three non-collinear points.
Congruent
When two segments have the same
measure they are said to be congruent
Symbol:
Example: A
B

>
<
<
>
D
C
AB CD
Midpoint / Segment Bisector
The midpoint of a segment is the point
that divides the segment into two
congruent segments
The Segment Bisector is a segment,
line or ray that intersects another
segment at its midpoint.
Example
Q is the Midpoint of PR, if PQ=6x-7 and
QR=5x+1, find x, PQ, QR, and PR.
Between
Point B is between point A and C if and
only if A, B and C are collinear and
AB  BC  AC
<
A
B
C
>
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
Example
Find the length XY in the figure shown.
Example
If S is between R and T and RS = 8y+4,
ST = 4y+8, and RT = 15y – 9. Find y.
Part 3
Angles
Angle
An angle is formed by two non-collinear
rays that have a common endpoint. The
rays are called sides of the angle, the
common endpoint is the vertex.
Kinds of angles
Right Angle
Acute Angle
Obtuse Angle
Straight Angle / Opposite Rays
Congruent Angles
Just like segments that have the same
measure are congruent, so are angles
that have the same measure.
Angle Addition Postulate
– If R is in the interior of <PQS, then
m<PQR + m<RQS = m<PQS
P
R
S
Q
– If m<PQR + m<RQS = m<PQS, then R is
in the interior of <PQS
Example
If m<BAC = 155, find m<CAT and
m<BAT
T
C
(3x+14) °
B
(4x-20) °
A
Example
<ABC is a straight angle, find x.
D
A
(11x-12) °
(2x+10) °
B
C
Angle Bisector
A ray that divides an angle into two
congruent angles is called an angle
bisector.
Example
Ray KM bisects <JKL, if m<JKL=72
what is the m<JKM?
Adjacent Angles
are two angles that lie in the same plane,
have a common vertex, and a common side,
but no common interior points
C
B
ADB is adjacent to
D
A
BDC
Vertical Angles
Two non-adjacent angles formed by two
intersecting lines
1
1 is vertical to
2
2
Vertical Angles have the same measure
and are congruent
Linear Pair
A pair of adjacent angles who are also
supplementary
1
2
1 and 2 are a liner pair
and m 1 + m 2 =180
Angle Relationships
Complementary Angles - Two angles
whose measures have a sum of 90
Supplementary Angles - are two angles
whose measures have a sum of 180
Examples
Part 3
Polygons
Polygon
Closed figure whose sides are all
segments.
– To be a Polygon 2 things must be true
• Sides have common endpoints and are not
collinear
• Sides intersect exactly two other sides
Examples
Non-Examples
Naming a Polygon
The sides of each angle in a polygon are the sides of
the polygon
The vertex of each angle is a vertex of the polygon
They are named using all the vertices in consecutive
order
Example
A
B
D
C
The number of sides determines the
name of the polygon
3 - Triangle
4 - Quadrilateral
5 - Pentagon
6 - Hexagon
7 - Heptagon
8 - Octagon
9 - Nonagon
10 - Decagon
12 - Dodecagon
Anything else …. N - gon (where n represents the
number of sides)
Concave VS Convex
Concave
Convex
Regular Polygon
A regular polygon is a convex polygon
whose sides are all congruent and
whose angles are all congruent
Perimeter
The perimeter of a polygon is the sum
of the lengths of its sides.
Example
l
s
a
b
s
s
w
w
s
c
p =a + b + c
l
p =s + s +s + s
p = 4s
p =l + w + l +w
p = 2l + 2w
Perimeter of the Coordinate Plane
Find the perimeter of the triangle ABC
with A(-5,1), B(-1,4), C(-6,-8)
Area
Area of a polygon is the number of
square units it encloses
h
h
b
b
1
A=
2
bh
A = bh
Circle
C = 2 šr
A = š r2
r
Unit 1
The End!