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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
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:
< >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:
< >
F
A BE
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. Name two points.2. Name two lines.3. Name two segments.4. Name two rays.
5. Name a line that does not contain point T.6. Name a ray with point R as the endpoint.7. Name a segment with points T and Q as its endpoints.8. Name three collinear points.9. Name three non-collinear points.
Congruent
When two segments have the same measure they are said to be congruent
Symbol:
Example:
≅
AB ≅ CD
< >
>< A B
C D
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
– If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS
P
Q
S
R
Example
If m<BAC = 155, find m<CAT and m<BAT
(3x+14)°
(4x-20)°B
A
TC
Example
<ABC is a straight angle, find x.
(2x+10)°(11x-12) °A
C
D
B
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
∠ADB is adjacent to ∠BDC
AD
C
B
Vertical Angles
Two non-adjacent angles formed by two intersecting lines
Vertical Angles have the same measure and are congruent
∠1 is vertical to ∠2
21
Linear Pair
A pair of adjacent angles who are also supplementary
∠1 and ∠2 are a liner pairand m∠1 + m∠2 =180
21
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 sidesNon-ExamplesExamples
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
D
C
B
The number of sides determines the name of the polygon
3 - Triangle 4 - Quadrilateral5 - Pentagon6 - Hexagon7 - Heptagon8 - Octagon9 - Nonagon10 - Decagon12 - DodecagonAnything else …. N - gon (where n represents the number of sides)
Concave VS Convex
ConvexConcave
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.
p = l + w + l + w
p = 2l + 2w
p = s + s + s + s p = 4s
p = a + b + c
ww
l
l
s
s
s
s
c
ba
Example
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
A = bhA =
1
2bh
h
b
h
b
Circle
C = 2πr
A = πr2
r
Unit 1
The End!