Download - Identify Points, Lines and Planes - Geometry
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Identify Points, Lines and Planes
“We can redeem anyone who strives
unceasingly.” -Goethe
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Identifying shapes
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Undefined terms
O Point-Has no dimension. It is represented by a dot.
O Line-Has one dimension. It is represented by a line
with two arrowheads, but continues forever. (note:
through any two points there is EXACTLY ONE line,
so we can name a line using only two points)
O Plane-Has two dimensions. It is represented by a
shape that looks like a wall or floor, but this also
continues forever in every direction.
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Other shapes
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Other Shapes
O Line segment-called a segment because it is
a finite part of the line. Labeled by its
endpoints.
O Ray-One side has endpoints but the other
extends forever. Labeled going from
endpoint to infinity.
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Ex. 1: Name all UNIQUE rays, lines, segments,
planes, and points.
A B C D
E F
G
H I J
R
S
K
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Intersection
O The set of point where two shapes meet.
O The intersection of two unique lines is
always a point.
O The intersection of two unique planes is
always a line.
m
n
A B
S
R
C
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Ex. 3: Sketch all interactions of a line and
plane. (there are three different ways)
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Ex. 3: Name the intersections of:
1)Plane ABD and Plane AEF
2)Line AD and Line HD
3)Segment EF and Segment GF
4)Plane EFH and Plane ABC
5)Segment AD and Segment HG
A
B C
D
E
F G
H
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Summary
O You should now be able to:
O Identify points, lines, segments, rays, and
planes.
O Identify points of intersection.
O Make sketches of each type of shape.
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Identify Angles and Angle Measures Part 1
“People don’t ever seem to realize that
doing what’s right is no guarantee against
misfortune.” –William McFee
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Parts of an angle
O Sides-line segments or rays that form the
angle.
O Vertex-the endpoint of the rays or the place
where the sides of an angle meet. (the
corner or sharp point of the angle)
O Naming angles
B
A
C
D E
F
G
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O Ex. 1: Name all the angles in the figure below.
A
B
C
D
E G
F
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Measuring with a Protractor
O Line up the protractor with a side and make
sure the vertex is in the center of the dot.
O Read off angle according to the type of
angle.
O Postulate 3: Protractor Postulate
O The measure of an angle is equal to the
absolute value of the difference between the
numbers for the sides.
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Identify Angles and Angle Measures Part 2
“People don’t ever seem to realize that
doing what’s right is no guarantee against
misfortune.” –William McFee
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Angle Addition Postulate
O Postulate 4: If P is in the interior of angle
RST then the measure of angle RST is equal
to the sum of the measures of angle RSP
and angle PST.
R
S
P
T
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O Ex. 2: Given that the measure of angle RST is 80°,
find the measure of angle PST and the measure of
angle RSP.
R
S
P
T
(3x – 5)°
(2x + 10)°
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Congruent and Bisectors
O Two angles are considered congruent if they have
the same measure. Congruent means that two
shapes are the same size and same shape. (like
the equal sign but for shapes instead of for
numbers). In Ex. 2, angle RSP is congruent to angle
PST.
O An angle bisector is a segment, ray, or line that
divides an angle into two smaller congruent angles.
In Ex. 2, ray SP ended up being an angle bisector of
angle RST.
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O Ex. 3: Ray AB bisects angle CAD. The measure of
angle BAD is 54°, what is the measure of angle
CAD and the measure of angle CAB?
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Summary
O You should know how to identify an angle.
O You should know how to measure an angle
using a protractor.
O You should know what a bisector does and
what congruent means.
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Segment Length and Congruence
“Our minds are lazier than our bodies.”
–La Rochefoucauld
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Ruler Postulate
O Postulate 1: The points on a line can be
matched one to one with the real numbers.
These numbers are called coordinates of the
point. The distance between points A and B,
written as AB, is the absolute value of the
difference of the coordinates of A and B.
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O Ex. 1: Measure the length of these segments to the
nearest tenth of a centimeter.
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Segment Addition Postulate
O Postulate 2: If B is between A and C, then
AB + BC = AC. The converse is true also. If
AB + BC = AC, then B is between A and C.
A B C
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Congruence
O Two segments are congruent if they have the
same length. If AB = BC, then segment AB is
congruent to segment BC.
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O Ex. 2: Use the diagram to find BC.
A B C
B A C
56
20
x
25 35
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O Ex. 3: Plot J(-2, 5), K(1, 5), L(3, 4), and M(3, -1) in
a coordinate plane. Then determine whether
segment JK and segment LM are congruent.
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Summary
O You should know what postulate 1 and 2
allow you do and understand that postulates
are rules that can’t be proved. These rules
are the foundations of geometry.
O You should understand the difference
between congruent and equal.
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Midpoint and Distance Formulas
“This is nothing wrong with making
mistakes. Just don’t respond with encores.
–Anon.
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Midpoint
O The midpoint of a segment is the point that
divides the segment into two smaller
congruent segments.
O A segment bisector is a shape that goes
through the midpoint of a segment.
A B M
C
D
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O Ex. 1: M is the midpoint of segment AB. If AM is
25, what is BM and AB?
O Ex. 2: M is the midpoint of segment AB. If AM is 2x-
5 and MB is x+3, what is AB?
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The Midpoint Formula
O The coordinates of the midpoint of a
segment are the averages of the coordinates
of the endpoints. If 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2)
are points on a coordinate plane, then the
midpoint M of segment AB has the
coordinates 𝑀(𝑥1+𝑥2
2,
𝑦1+𝑦2
2).
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O Ex. 3: Find the coordinates of the midpoint M of
segment AB if A(1, 2) and B(5, -4).
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O Ex. 4: Find the coordinates of the endpoint C of
segment CD. The midpoint of segment CD is N(1,4)
and the endpoint D(3, 5).
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Distance Formula
O If 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2) are points on a
coordinate plane, then the distance between
A and B is 𝐴𝐵 = (𝑥2 − 𝑥1)2+(𝑦2 − 𝑦1)2.
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O Ex. 5: What is the length of segment AB with
endpoints A(1, 2) and B(-3, -1)?
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Summary
O You should be able to use the midpoint
formula to find the coordinates of an
endpoint or midpoint of a segment.
O You should be able to use the distance
formula to find the length of segment.
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Angle Pairs “Being entirely honest with oneself is a
good practice” –Sigmund Freud
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Complementary and Supplementary Angles
O Complementary angles are two angles that
measures sum to 90°.
O Supplementary angles are two angles that
measures sum to 180°.
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O Ex. 1: Identify/Name the supplementary angles, the
complementary angles and the adjacent angles.
A
B
C
D E
F
G
22°
112° 68°
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O Ex. 2: Given that angle 1 and angle 2 are
complements and measure of angle 1 is 78°, find
measure of angle 2.
O Ex. 3: Given that angle 3 and angle 4 are
supplements and measure of angle 4 is 112°,
find the measure of angle 3.
O Additional question: Which angles given above are
adjacent?
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Linear Pair
O Linear Pair-Two adjacent angles with their
noncommon sides forming opposite rays.
(could be though of two adjacent
supplementary angles)
1 2
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Vertical Angles
O Two angles with sides forming two pairs of
opposite rays
1 2
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O Ex. 4: Identify all vertical angles and linear pairs.
1
2 3
4
5
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O Ex. 5: Two angles form a linear pair and one angle
is 3 times the measure of the other angle. What
are the measures of the angles?
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O What can you conclude.
A B C
D E
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Summary
O You should be able to identify
complementary and supplementary angles.
O You should be able to identify linear pairs
and vertical angles.
O You should be able to use the above
definitions to solve a problem.
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Classify Polygons “Experience is a good teacher, but she
sends in terrific bills.”
-Minna Antrim
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Polygons
O Poly-multi and gons-sides. Translates to
multisided shapes.
O Two requirements:
O It is formed of three or more segments called
sides.
O Each side intersects EXACTLY two sides, one
at each endpoint, so that no two sides with a
common endpoint are collinear.
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Convex and Concave
O If you could draw a line from any two
vertices of a polygon and that line would
only be in the interior or the side of polygon,
then that polygon is convex.
O If at least one line drawn from two vertices
of a polygon lies in the exterior, then the
polygon is concave.
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Classifying Polygons Number of Sides Type of Polygon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
11 Hendecagon
12 Dodecagon
n N-gon
100 100-gon
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Special Polygons
O Equilateral - All sides are congruent (all
sides have the same length)
O Equiangular - All angles are congruent (all
angles have the same measure)
O Regular – Polygon is both equiangular and
equilateral
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Ex. 1: Tell whether each shape is a polygon. If it is a
polygon, classify the polygon and state whether it is
concave or convex.
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Ex. 2: Classify each polygon and tell whether the
polygon is equilateral, equiangular, regular, or none of
these. Explain.
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Ex. 3: A stop sign is a regular octagon. The
expressions for two of the side lengths are (4x – 4)in.
and (3x + 6)in. What is the length of the edge of the
stop sign?
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Perimeter, Circumference, and Area of Basic Shapes
“To travel hopefully is a better thing than to
arrive, and the true success is to labour.”
-Robert Louis Stevenson
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Terms
O Perimeter (P) – the distance around a
polygon
O Circumference (C)– the distance around a
circle
O Area (A) – the amount of surface covered by
the figure.
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Formulas
Square
𝑃 = 4𝑠
𝐴 = 𝑠2
Side length s
Rectangle
𝑃 = 2𝑙 + 2𝑤
𝐴 = 𝑙𝑤
Length l and width w
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Formulas
Triangle
𝑃 = 𝑎 + 𝑏 + 𝑐
𝐴 =1
2𝑏ℎ
Side lengths a, b, and c;
base b; and height h
Circle
𝐶 = 𝜋𝑑 𝑜𝑟 𝐶 = 2𝜋𝑟
𝐴 = 𝜋𝑟2
Diameter d and radius r
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Ex. 1: Find the perimeter and area of a football field if
it has a width of 50 yards and length of 120 yards.
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Ex. 2: Find the area and perimeter of a triangle with
the given dimension below
3.5
2 2 1
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Ex. 3: Triangle ABC has coordinates A(1, 6), B(1, 2),
and C(7, 3). Find its perimeter and area.
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Ex. 4: Find the circumference and area with a
diameter of 10 in. and a center C(5, 7).
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Summary
O Students should be able to find the
perimeter and area of a triangle, rectangle
and square.
O Students should be able to find the
circumference and area of a circle.
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Bisect Segments and Angles
“The difference between what we do and what we are capable of doing would suffice to solve most of the world’s
problems.” -Gandhi
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No Measuring Device
O Without a ruler or protractor we can
construct squares, perpendicular lines,
equilateral triangles, bisect angles perfectly,
etc.