Download Geometry Unit 1

Point - A point has no dimension. It is represented
Line - A line has one dimension. It is represented
by a dot. It represents a fixed point in space.
by a line with two arrowheads, but it extends without
end.
Through any two points,
there is EXACTLY one line.
You can use any two points
on a line to name it.
Plane - A plane has two dimensions. It is
Collinear – Points that lie on the same line.
represented by a shape that looks like a floor or a
wall, but it extends without end.
Through any three points
not on the same line, there
is exactly one plane.
You can use three points
that are not all on the same
line to name a plane.
Coplanar - Points that lie in the same plane.
Points S, P, and T are
collinear.
Points P and Q are
also collinear.
Endpoint – A point that is the beginning or end of a
segment or ray.
Points S, P, T and V
are coplanar. They
all lie in plane R.
Segment – A piece of a line which consists of two
Ray – A piece of a line which consists of one
endpoints.
endpoint and extends indefinitely in the other
direction.
Written as 𝑨𝑩 or 𝑩𝑨
⃗⃗⃗⃗⃗ starts a point A
𝐴𝐵
and runs through point B
⃗⃗⃗⃗⃗ starts a point B
𝐵𝐴
and runs through point A
⃗⃗⃗⃗⃗ and 𝐴𝐵
⃗⃗⃗⃗⃗ are different rays.
So 𝐵𝐴
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Opposite Rays – Two rays that share a common
Theorem - A rule that can be proved.
endpoint and extend in opposite directions.
⃗⃗⃗⃗⃗ and CB
⃗⃗⃗⃗⃗ 𝑎𝑟𝑒
CA
opposite rays
Postulate – A rule that is accepted without proof.
(also called axiom)
Distance – The distance between points A and B,
written as AB, is the absolute value of the difference
of the coordinates of A and B.
Distance of a segment is written as:
⃗⃗⃗⃗⃗
AB or the length of 𝐴𝐵
Congruent Segments – Line segments that have
Construction – A geometric drawing that uses a
the same length.
limited set of tools, usually and compass and a
straightedge (a ruler without marks)
or AB = CD
Angle – Two different rays with the same endpoint.
This angle can be named
BAC
CAB
A
Point A is the vertex of the angle.
(Segment) Bisector – A point, ray, line, line
segment, or plane that intersects the segment at its
midpoint.
Interior/Exterior of an Angle – A point is in the
Vertex – The common endpoint of two different rays
interior of an angle if it is between points that lie on
each side of the angle.
that form an angle.
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Congruent Angles –
Vertical Angles – Two angles whose sides form
Two angles are congruent
if they have the same measure.
two pairs of opposite rays.
Adjacent Angles – Two angles that share a
Acute/Right/Obtuse/Straight
common vertex and side, but have no common
interior points.
Linear Pair – Two adjacent
Non-Euclidean Geometry – Geometry that is not
angles are a linear pair if
their noncommon sides are
opposite rays. The angles
in a linear pair are
supplementary angles.
based on a flat surface.
Euclidian Geometry – aka “flat” or “parabolic”
Complementary/Supplementary
geometry. Named after the greek mathematician
Euclid.
Geometry based on the assumption of a flat surface.
Complementary angles – the sum of their measures is
90°
Supplementary angles – the sum of their measures is
180°
Some types of non-euclidean geometry:
Hyperbolic Geometry
Elliptic Geometry
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1.1 Vocabulary Grid
Write down a definition, example or sketch a picture of each word. After you have filled in as many as
possible walk around the room and find a different person for each box to help you fill it in. As you fill
out others sheet give an educated guess if you do not know a term. You have 8 minutes to get all the
boxes filled.
Point
Intersection
Segment
Endpoint
Line
Coplanar
Plane
Ray
Defined Term
Opposite Ray
Undefined Term
Collinear
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Section 1.1 – Points, Lines, and Planes
Ordered Pair (x,y)
Quadrant II
Quadrant I
Quadrant III
Quadrant IV
x-coordinate _________________
y-coordinate_________________
Collinear Points: points that lie on the same line
Name three collinear points:
Name three points that are not collinear:
Ex 1: Find the coordinates of three points that lie on the graph of y=3x+5 and one point that doesn’t lie on the
line
x
3x + 5
y
Ex 2: Do points R (4,8) and S (-3,-7) lie on the line with the equation y=2x-1?
Undefined Terms – No formal definition, but it is understood. The terms point, line, and plane are undefined terms.
Point: An exact position in space; Represented by a dot
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Line: Collection of points along a straight path extending infinitely in opposite directions.
Plane: Flat surface extending infinitely in all directions.
Coplanar: Points that lie in the same plane.
Noncoplanar: Points that do not lie in the same plane.
Practice:
Defined Terms: Terms that can be described using known words such as point or line.
Line Segment: Part of a line consisting of two endpoints and all points on the line between the
endpoints.
Ray: Part of a line with one endpoint that extends infinitely in the opposite direction.
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Practice:
Intersections – Two or more geometric figures intersect if they have or more points in common.
Perpendicular Lines: Two lines that intersect to form a right angle. (opposite reciprocal slope)
Parallel Lines: Two coplanar lines that never intersect. (same slope)
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Section 1.2 – Segments & Congruence
The length of a segment is the distance between its two endpoints.
The length of ST can be written as ________
Postulate/Axiom – A rule that is accepted without proof
Ruler Postulate: The numbers on a ruler are a real-life example of a number line. The distance
between two points on a number line is the absolute value of the difference of the coordinates.
Example 1: What is the distance from X to Y?
X
Y
Distance from X to Y can be written as ________ or __________
When three points are collinear, you can say that one point is between the two others.
Example 2: Find PQ, QR, and PR if P is located at -3, Q is located at 1, and R is located at 6.
Segment Addition Postulate:
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Example 3: Find LM if L is between N and M, NL = 6x-5, LM = 2x+3, and NM = 30
Example 4: Find the length of MN if N is between M and P, MN=3x+2, NP=18, MP=5x
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Congruent – Same value
Congruent Segments: Line segments that have the same length.
Example 5: Find AB if K is between A and B, AK=2x+10, KB=5x+4, and AK is congruent to KB.
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You will need a compass and a ruler to do the following.
Use the following segments to connect them on the dotted line provided:
M
A
B
B
R
C
D
T
D
A
̅̅̅̅ and 𝐵𝐶
̅̅̅̅ : AB=, BC= , AC=___+_____=____
Add segments 𝐴𝐵
●
̅̅̅̅ and 𝐷𝑀
̅̅̅̅̅: AD=, DM= , AM=___+___=___
Add segments 𝐴𝐷
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●
Section 1.3 – Midpoint and Distance Formulas (Day 1)
Midpoint (of a segment) – The point that divides the segment into two congruent segments.
Segment Bisector – A point, ray, line, line segment, or plane that intersects the segment at its midpoint.
To determine the midpoint on a number line, add the two endpoints and divide by 2.
Midpoint Formula:
𝒙𝟏 + 𝒙𝟐
𝟐
Example 1 – Midpoint
X
A
Y
B
What is the coordinate of the midpoint of segment AB?
What is the coordinate of the midpoint of segment XY?
Example 2 – Midpoint: Line RM bisects segment LK at point D. Find DK if LK is 16.
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Example 3 - Midpoint: Point M is the midpoint of segment VW. Find the length of segment VM.
Example 4 - Midpoint: If P is the midpoint of segment CD, PC=x and PD=5x-4, find the value of x and the length
of CD.
Coordinate Plane Midpoint Formula:
Example 5 – Coordinate Plane Midpoint: The endpoint of segment RS are R(1,-3) and S(4,2). Find the
midpoint.
Example 6 – Missing Endpoint: The midpoint of segment JK is M(2,1). One endpoint is J(1,4). Find the
coordinates of endpoint K.
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Coordinate Plane Distance Formula
Example 7 - Distance: Find the length of RS if R(1,-3) and S(4,2).
Example 8 - Distance: Determine if the two segments AB and LK are congruent.
A(4,6) B(7,2)
L(-1,-6) K(4,-6)
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1.3 – Using the Midpoint and Distance Formulas (Day 2)
Midpoint Formula
Distance Formula
The coordinates of the midpoint of a segment
are the averages of the x-coordinates and of
the y-coordinates of the endpoints.
Find the midpoint and the distance for the following segments:
1. (2,3) and (2, 7)
2. (1,5) and (5,5)
Find the midpoint and the distance for the following segments:
4. (4,5) and (9,17)
5. (2,8) and (6, 4)
3. (2, 4) and (4,8)
6. (2,3) and (11,16)
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Given the following endpoint and midpoint, find the missing endpoint and the distance:
7. Endpoint: (4,-4)
8. Endpoint: (-3, 0)
9. Endpoint: (4, 4)
Midpoint: (1,-1)
Midpoint: (0, 1)
Midpoint (-1, -2)
Endpoint_____________
Endpoint_____________
Endpoint_____________
Distance______________
Distance______________
Distance______________
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Section 1.4 – Measure and Classify Angles
An ___________ is formed by two different rays with the same endpoint.
Name angles with three letters. The vertex must be in the middle.
____________________________
____________________________
___________________________
None of the angles should be named < X, because all three angles have X as their vertex.
Angles are measured in __________.
If an angle is 32 degrees, it can be written two different ways:
Angles are measured with a protractor. The image
to the right demonstrates how a protractor is used.
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Classifying Angles
Angle Addition Postulate
Example 1 – Find Angle Measures:
Given that
find
Example 2 - Find Angle Measures:
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Congruent Angles – Angles that have the same measure.
Angle Bisector – A ray that divides and angle into to congruent angles.
Since < XYW and < ZYW are congruent, ray YW is the angle bisector of < XYZ.
Example 3 – Double an Angle Measure:
Example 4 – Challenge
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http://www.mathopenref.com/constcopysegment.html
Use the steps and copy the following segment:
http://www.mathopenref.com/constcopyangle.html
Use the steps above and copy the following angle right next to it not under:
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Section 1.5 – Angle Pair Relationships
Adjacent Angles – Two angles that share a common vertex and side.
Complementary Angles – Two angles whose sum is 90°
Supplementary Angles – Two angles whose sum is 180°
Sum is 90°
Sum is 180°
Example 1: Find the measure of the complement and supplement of
m 1 = 22°
1.
Example 2: What are the measures of angles 2 and 3?
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Example 3:
Linear Pair
Vertical Angles
Vertical angles are always congruent!!
Example 4:
m
FGH and
FHG=(12x+1)° and m
HGJ form a linear pair. Find the measures of the angles if
HGJ=(4x-9) °. (Draw a picture to help you)
Example 5: Find the values of x and y.
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1.5 Class Practice
1) Tell whether the angles are supplementary, complementary, vertical, bisecting. For each identify if adjacent or
not.
2) Find m2=____ , 3=_______
3)
Solve for x and y:
Equation:
Equation:
Equation:
4) Use one of the learned concepts ( Complementary, supplementary, vertical, angle addition), set up an equation
and solve it.
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Section 1.6 – Classify Polygons
Polygon – A closed plane figure with the following properties:
1. It is formed by three or more sides (have to be straight)
2. No lines in the interior of the shape
A polygon can be named
by listing the vertices in
consecutive order. Two
possible names for this
polygon are ABCDE and
CDEAB. DBCEA would
not be a name for this
polygon because the
vertices are not listed in
the order they occur on
the polygon.
Vertex
(plural: vertices)
Convex and Concave
All interior
angles are
At least
one
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Example 1:
Naming Polygons – A polygon is named by the number of sides.
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Equilateral – All sides congruent
Equiangular – All angles congruent
Regular – both equilateral and equiangular
Example 2:
Example 3: Find the value of x in the regular polygon.
Section 1.7 – Find Perimeter, Circumference and Area
Perimeter: Distance around a figure
Circumference: Distance around a circle
Area: Amount of surface covered by a figure
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Example 1: Find the perimeter and area of the rectangular basketball court shown.
Example 2: Find the approximate circumference and area of the patch shown.
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Example 3: The base of a triangle is 28 meters. Its area is 308 square meters. Find the
height of the triangle.
Example 4: Find the area and perimeter of the triangle.
(Use distance formula to find the length of each side)
1.7 Class Practice
1)
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Use the distance (length) formula to find out the length of each one QR, QS, and RS
QR=√
QS=√
RS=√
Perimeter=
2) Read this solved problem and solve the next one in a similar way:
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h=_____________________
3)
a) Circumference rend ring=________________
Circumference yellow ring=________________
b) _______________________________________
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