1.1 Building Blocks of Geometry

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1.1 Building Blocks of Geometry
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Geometry Terms

• Definition Known words used to describe a new
  word.
• Postulate A statement that is accepted as true
  without proof sometimes they are called axioms.
• Theorem Important statements that are proven.

Homework
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Segment

1. Begins at one point and ends at another
2. Has points on each end called endpoints
3. Consists of an infinite amount of points
4. Always straight
5. Named by its endpoints, in either order

• Can be called
• Segment AB
• Segment BA
• AB or BA

A
B
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Point

• 1. Has no dimension
• (no length, width, thickness)
• 2. Usually represented by a dot
• 3. Named using one capital letter
• B

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Line

1. Extends forever in one dimension (length)
2. Has an arrowhead on each end representing the
   fact that it goes on forever
3. Consists of an infinite amount of points
4. Always straight
5. Named with a lowercase script letter or by two
   points on the line

Can be called Line l Line AB or AB Line BA or BA
l
A
B
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Plane

1. Extends forever in 2 dimensions (length width)
2. A flat surface consisting of infinitely many
   points
3. Usually represented by a 4-sided figure
4. Named with a capital script letter or 3
   noncollinear points on the surface of the plane

Can be named Plane W Plane ABC, plane BCA, plane
CBA, (any three noncollinear points)
A
W
C
B
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Different planes in a figure
A
B
Plane ABCD Plane EFGH Plane BCGF Plane
ADHE Plane ABFE Plane CDHG
C
D
E
F
H
G
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Other planes in the same figure

• Any three non collinear points determine a plane!

Plane AFGD Plane ACGE Plane ACH Plane AGF Plane
BDG
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More Definitions

• Collinear points points that lie on the same
  line

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More Definitions
Coplanar points points that lie on the same
plane Coplanar lines lines that lie on the same
plane
A, B, and C are coplanar points Lines l and n are
coplanar lines
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Is Alex between Ty and Josh?
Yes!
Ty
Alex
Josh
No, but why not?
How about now?

• In order for a point to be between two others,
  all 3 points MUST BE collinear!!

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Ray

• Piece of a line with only one endpoint (initial
  point) and continues forever in the other
  direction
• Named by the endpoint and a second point named on
  the ray. (name MUST begin with the endpoint!)
• AB

A
B
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Opposite Rays

• Two rays that share a common initial point and
  face opposite directions.
• QP and QS are opposite rays.

P
Q
S
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More Definitions

• Intersect two or more figures intersect if they
  have one or more points in common.
• Intersection all points or sets of points the
  figures
• have in common
• What is the intersection of
• AB DA
• BC AC
• BC BC

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When two lines intersect, their intersection is a
point.
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When two planes intersect, their intersection is
a line.
B
P
A
R
Plane P and Plane R intersect at the line
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Angle symbol

• Two rays that share the same endpoint (or initial
  point)

Sides the rays XY XZ Vertex the common
endpoint X
Y
X
5
Z
Angles can also be named by a . (?5)
Named ?YXZ, ?ZXY (vertex is always in the
middle), or ?X (if its the only ?X in the
diagram).
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There are 3 different ?Bs in this diagram
therefore, none of them should be called ?B.
A
?B?
D
B
C
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Interior or Exterior?

• B is ___________
• C is ___________
• D is ___________

in the interior
in the exterior
on the ?A
B
C
D
A
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AssignmentSection 9 - 47
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1.2 Measuring Length
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Ruler postulate

• The points on a line can be matched with those on
  the real number line.
• The real number that corresponds to a point is
  the coordinate of the point.
• If you find the difference between the
  coordinates of two points, then take the absolute
  value, you will have the distance or length
  between the points.

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Ruler postulate (continued)
A
B

• The symbol for the length of AB is AB.

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Example Find AB.
A
B
Point A is at 1.5 and B is at 5. So, AB 5 -
1.5 3.5
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• Determine the length of a given segment.

AB ? 4 (1)? 3
BC ?1 4? 5
CD ?4 9? 5
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• Determine whether segments are congruent.

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Segment Congruence Postulate
Segment Congruence Postulate If two segments
have the same length then the segments are
congruent. Also if two segments are congruent
then they have the same length if measured by a
fair ruler.

• If AB XY have the same length,
• Then AB XY,
• and
• AB XY

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Segment Addition Postulate
If B is between A C, then AB BC AC. If AB
BC AC, then B is between A C.
C
B
A
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Example If DE 2, EF 5, and DE FG, find
FG, DF, DG, EG.
D
E
F
G
FG 2 DF 7 DG 9 EG 7
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Questions
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AssignmentPractice A, B and Section 11 - 27
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1.3 Measuring Angles
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• A protractor is a device used for measuring
  angles. As on a ruler the intervals on a
  protractor are equal.

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Angle Measurement

• m?A means the measure of ?A
• Measure angles with a protractor.
• Units of angle measurement are degrees (o).

• Angles with the same measure are congruent
  angles.
• If m?A m?B,
• then ?A ?B.

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Measure of an Angle

• The rays of an angle can be matched up with real
  numbers
• (from 0 to 180) on a protractor so that the
  measure of the ? equals the absolute value of the
  difference of the numbers.

55o
20o
m?A 55 - 20 35o
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• Determine the measure of a given angle.

Find the measures of angle ?BVC.
m ?BVC ?125? 50?? 75?
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• Add measures of angles.

m ?AVC m ?AVB m ?BVC
25? 75? 100?
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Angle Addition Postulate

• If P is in the interior of ?RST,
• then m?QRP m?PRS m?QRS.

If m?QRP 5xo, m?PRS 2xo, m?QRS 84o,
find x. 5x 2x 84 7x 84 x
12 m?QRP 60o m?PRS24o
S
P
Q
R
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Angle Congruence Postulate
If two angles have the same measure, then they
are congruent. If two angles are congruent, then
they have the same measure.
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Types of Angles

• Acute angle
• Right angle
• Obtuse angle
• Straight angle

• Measures between 0o 90o
• Measures exactly 90o
• Measures between 90o 180o
• Measures exactly 180o

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Adjacent Angles

• Two angles that share a common vertex side, but
  have no common interior parts.
• (they have the same vertex, but dont overlap)
  such as ?1 ?2

2
1
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Example

• Name an acute angle
• ?3, ?2, ?SBT, or ?TBC
• Name an obtuse angle
• ?ABT
• Name a right angle
• ?1, ?ABS, or ?SBC
• Name a straight angle
• ?ABC

S
T
3
1
2
A
B
C
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Midpoint

• The point that bisects a segment.
• Bisects?
• splits into 2 equal pieces

12x 3 10x 5
12x 3 10x 5 2x 2 x 1
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Segment Bisector

• A segment, ray, line, or plane that
• intersects a segment at its midpoint.

k
A
M
B
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Angle Bisector

• A ray that divides an angle into two congruent
  adjacent angles.
• BD is an angle bisector of ?ABC.

A
D
B
C
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Example If FH bisects ?EFG and m?EFG 120o,
then what is m?EFH?
E
H
F
G
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Example Solve for x.
If they are congruent, set them equal to each
other, then solve!
x 40 3x - 20 40 2x - 20 60 2x
30 x
x 40o
3x - 20o
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Which angles are adjacent?
?1 ?2, ?2 ?3, ?3 ?4, ?4 ?1
Then what do we call ?1 ?3?
Vertical Angles two angles that share a common
vertex whose sides form 2 pairs of opposite
rays. ?1 ?3, ?2 ?4
2
1 3
4
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Linear Pair

• A linear pair is two adjacent angles whose
  non-common sides are opposite rays.
• These angles form a straight line and their sum
  is 180.

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Example

• Vertical angles?
• ?1 ?4
• Adjacent angles?
• ?1 ?2, ?2 ?3,
• ?3 ?4, ?4 ?5, ?5 ?1
• Linear pair?
• ?5 ?4, ?1 ?5
• Adjacent angles not a linear pair?
• ?1 ?2, ?2 ?3, ?3 ?4

2
1 3
5 4
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Important Facts

• Vertical Angles are congruent.
• The sum of the measures of the angles in a linear
  pair is 180o.

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Example

• If m?5 130o, find
• m?3
• m?6
• m?4

4
130o 50o 50o
5 3
6
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Example
A
E
3x 5o y 20o
B
x 15o 4y - 15o

• Find x and y
• m?ABE
• m?ABD
• m?DBC
• m?EBC

D
C
x 40 y 35 m?ABE 125o m?ABD 55o m?DBC
125o m?EBC 55o
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Complementary Angles

• Two angles whose sum is 90o

35o
1
2
55o
A
?1 ?2 are complementary ?A ?B are
complementary
B
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Supplementary Angles

• Two angles whose sum is 180o

?1 ?2 are supplementary. ?X ?Y are
supplementary.
1 2
130o 50o
X Y
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Example ?A ?B are supplementary. m?A is 5
times m?B. Find m?A m?B.

• m?A m?B 180o
• m?A 5(m?B)
• Now substitute!
• 5(m?B) m?B 180o
• 6(m?B)180o
• m?B30o
• m?A150o

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Perpendicular Bisector
A perpendicular bisector intersects at the
midpoint AND is perpendicular to the segment.
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Parallel Lines
Parallel Lines Two lines are parallel lines if
they lie in the same plane and do not intersect.
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Perpendicular Lines
Two lines are perpendicular lines if they
intersect to form right angles.
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Skew Lines

• Skew are lines that do NOT lie in the same plane
  and do NOT intersect.

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Questions
Find the measure of each of the angles.
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Questions
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Questions
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Questions
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AssignmentPractice B and Section 14 - 44