Geometry Chapter 1 Section 1.1 Basics Section 1.2 Angle

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Geometry Chapter 1

• Section 1.1 Basics
• Section 1.2 Angle Structures
• Section 1.3 Types of Angles
• Section 1.4 Polygon Basics
• Section 1.5 Triangles and Quadrilaterals
• Section 1.6 Circle Basics

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Section 1.1 Basics

Point
A location in space.
Line
A set of collinear points having no endpoints.
Plane
A set of points that creates a flat surface that
has no depth or edge.
Collinear
A
B
A set of points that lie on a line.
A
Coplanar
A set of points that lie on a plane.
B
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Section 1.1 Basics
A set of collinear points having two endpoints.
Line Segment
A
B
Congruent Segments
Two or more line segments that are identical
X
Y
A point that divides bisects a line segment
into two equal parts.
Midpoint of a Segment

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Section 1.1 Basics
Midpoint of a Segment (formula)
x 1, y1
x 2, y2
x 1 x2 , y 1 y2 2
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Ray
A set of collinear points having one endpoint.
Bisector of a Segment
A segment, line, or ray that intersects a segment
at its midpoint.
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Section 1.1 Basics
Homework Assignment Pages 33 34 Problems
2,3,5,8,9,10,11,18,19,20,24,27,28 Page
37 Problems 1 - 7
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Section 1.2 Angle Structures
Angle
The union of two noncollinear rays having a
common endpoint.
XA and XB are the sides of the angle and point X
is the vertex of the angle.
Congruent Angles
Two or more angles with the same measure
?AXB ? ?RST means that the measure of both angles
are equal.
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Section 1.2 Angle Structures
Angle of Reflection
The reflection of an angle will have the same
measure from the reflected surface as the initial
angle.
50o
40o
40o
50o
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Section 1.2 Angle Structures
Homework Assignment Pages 42 44 Problems
1,2,4,5,7 14, 30 - 35
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Section 1.3 Types of Angles
An angle whose measure is equal 90 o.
Right Angle
Acute Angle
An angle whose measure is lt 90 o.
Obtuse Angle
An angle whose measure is gt 90 o.
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Section 1.3 Types of Angles
Complimentary Angles
Two angles whose sum of measure equals 90. ?1 ?
2 90.
Two angles whose sum of measure equals 180. ? 1
? 2 180.
Supplementary Angles
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Section 1.3 Types of Angles
Opposite and congruent angles formed by the
intersection of two lines.
Vertical Angles
Two adjacent angles whose exterior sides lie in a
line.
Linear Pair of Angles
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Section 1.3 Types of Angles
Homework pages 51 53 Problems 1 8, 11 20,
(honors 25 30)
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Section 1.4 Polygon Basics
Polygon
A closed plane figure formed by the connecting
line segments endpoint to endpoint with each
segment intersecting exactly two other segments.
A polygon having the fewest possible number of
sides.
Triangle
n-gon
A polygon with n number of sides where n is equal
to any positive integer.
Octagon
Triangle
n 3
n 8
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Section 1.4 Polygon Basics
Naming Polygons
n 5
Pentagon
Hexagon
n 6
Octagon
n 8
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Section 1.4 Polygon Basics
Polygon Features
Consecutive angles
?FAB ?ABC
Consecutive sides
Consecutive vertices
?F ?E
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Section 1.4 Polygon Basics
Diagonal of a Polygon
A line segment that connects two nonconsecutive
vertices of a polygon.
AC. AD, AE are each diagonals of this polygon.
diagonals n(n-3) 2 n the of sides of
the polygon.
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Section 1.4 Polygon Basics
Congruent Polygons
If polygons are congruent then corresponding
sides and angles are congruent. Also, if
corresponding sides and angles of a polygon are
congruent then the polygons are congruent.
If ?ABC ? ?XYZ, then ?A ? X, AB XY ? B ?
Y, BC YZ ? C ? Z, AC XZ
?
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Section 1.4 Polygon Basics
Groups of Polygons
A polygon with all sides equal in length.
Equilateral polygon
A polygon with all angles equal in measure.
Equiangular polygon
A polygon with all angles equal in measure and
all sides equal in length.
Regular polygon
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Section 1.4 Polygon Basics
Groups of Polygons
Equilateral polygon
The rhombus has all sides congruent.
The rectangle has all angles congruent.
Equiangular polygon
The square has all angles and sides congruent.
Regular polygon
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Section 1.5 Triangles and Quadrilaterals
Right Triangle
A triangle with one right angle.
Acute Triangle
A triangle with three acute angles.
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Section 1.5 Triangles and Quadrilaterals
Obtuse Triangle
A triangle with one obtuse angle.
Scalene Triangle
A triangle with no congruent sides.
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Section 1.5 Triangles and Quadrilaterals
Equilateral Triangle
A triangle with all sides congruent.
Isosceles Triangle
A triangle with two congruent sides.
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Section 1.5 Triangles and Quadrilaterals
Trapezoid
A quadrilateral with two sides parallel.
B
A
C
D
A quadrilateral with both pairs of opposites
sides parallel and congruent.
Parallelogram
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Section 1.5 Triangles and Quadrilaterals
Kite
A quadrilateral with two pair of adjacent sides
congruent.
A quadrilateral with opposite sides parallel and
all sides congruent.
Rhombus
A
B
D
C
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Section 1.5 Triangles and Quadrilaterals
Rectangle
A quadrilateral with opposite sides parallel and
congruent with adjacent sides perpendicular.
A
B
D
C
A quadrilateral with opposite sides parallel ,
adjacent sides perpendicular, and all sides
congruent.
Square
A
B
D
C
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Section 1.5 Triangles and Quadrilaterals
Homework Pages 64 66 1 20, 25 - 29
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Section 1.6 Circles

• Circle The set of points equidistant to a given
  point.
• Radius The distance from the center to the edge
  of a circle. XA
• Diameter The distance through the center from
  opposite edges. EB
• Chord A segment whose endpoints lie on the
  circle. FD
• Tangent A line, segment, or ray that intersects
  a circle at only one point. Line C
• Arc The distance between two points on a
  circle. AB

A
F
B
E
X
D
C
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Section 1.6 Circles

• Major Arc An arc whose measure is gt 180 o. FAC,
  ABD, EFD
• Minor Arc An arc whose measure is lt 180 o. FA,
  AB, EF, BD
• Semicircle An arc whose measure is equal to 180
  o . EAB
• Central Angle An angle inside of a circle whose
  vertex is the center of the circle. ? CXB, ? AXB

A
F
B
E
X
D
C
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Section 1.6 Circles
Homework pages 70 72 1 8, 10, 21 24,
(honors only 25 33)
Test in two days on Chapter 1
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Introduction to Geometric Proofs

• Proofs use given information, diagrams,
  definitions, postulates, and theorems to prove a
  suggested conclusion.
• Proofs develop reasoning skills and analysis of
  circumstances in a math application
• Mathematical proof in all math subjects are used
  to illustrate relationships between the
  conditions of a problem and its solution.

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Properties of Equality

• Reflexive a a
• Symmetric If a b, then b a
• Transitive If a b and b c, then a c.
• Addition If a b, then a c b c
• Subtraction - If a b, then a - c b c
• Multiplication - If a b, then a c b c
• Division If a b and c? 0, then a/c b/c

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Properties of Equality

• Substitution Any value that is congruent to
  another value can be substituted for the given
  value.
• Example Let a 5 if a b c then 5 b c
• Example Let ?A ? ?B and ?B ?C 180 then we
  can say ?A ?C 180

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Postulates

• Postulates are mathematical statements that are
  accepted as true and obvious without need of
  formal proof.
• A line contains at least two points, a plane
  contains at least 3 points not in a line, space
  contains at least four points not all in one
  plane.

1 dimension
2 dimensions
3 dimensions
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Postulates

• Through any two points there is exactly one line.
• A B
• Through any three points there is at least one
  plane and through any three non-collinear points
  there is exactly one plane.
• . A B .
• . C

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Postulates

• If two points are in a plane then the line that
  contains the points is in the plane.
• A B
• If two planes intersect, then their intersection
  is a line.

If points A and B are in the plane then line AB
is in the plane.
B
A
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Theorems

• Vertical angles are congruent.
• ?1 ??2.
• Adjacent angles formed by ? lines are congruent.
• With AB ? to XY, we create two right angles,
• ? AXY and ?BXY. Since right angles
• Always equal 90 degrees we can conclude
• That ? AXY ??BXY.

2
1
Y
A
B
X
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Theorems

• If 2 lines form ? adjacent angles they are ?.
• With ? AXY ??BXY then can conclude that AB is ?
  to XY
• If the exterior sides of two adjacent angles
• are ? then the angles are complementary

With XA ? XC, and ?AXB adjacent to ?BXC, we can
conclude that ?AXC is a right ? and ?AXB ?BXC
90.
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Theorems

• If two angles are supplements to congruent angles
  or the same angle, then the angles are congruent.
• If two angles are complements to congruent angles
  or the same angle, then the angles are congruent.

1
2
3
4
?2 ??4, ?1 is supplementary to?2, ?3 is
supplementary to?4 therefore ?1 ??3.
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Theorems

• If two angles are complements to congruent angles
  or the same angle, then the angles are congruent.

?2 ??4, ?1 is complementary to?2, ?3 is
complementary to?4 therefore ?1 ??3.
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Theorems

• If two lines intersect, then they intersect in
  exactly one point.

Line l intersects line m at A.

• If there is a line and a point not on the line,
  then exactly one plane contains them both.

. A
With line xy and point A not intersecting each
other, there is only one possible plane that
contains them both.
y
x
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Theorems

• If two lines intersect, then exactly one plane
  contains them.

• If M is the midpoint of AB, then 2AM AB and AM
  ½ AB BM ½ AB and 2BM AB

.
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Two Column Proofs

• Given P lies between A and B.
• Prove PB AB - AP

.
.
.
Statement
Reason

• P lies between A and B
• AP PB AB
• PB AB - AP

• Given
• Betweeness of Points
• Subtraction Property of

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Two Column Proofs

• Given AB CD
• Prove AC BD

.
.
.
.
A
D
B
C
Statement
Reason

• AB CD
• AB BC AC
• CD BC BD
• BC BC
• AB BC CD BC
• AC BD

• Given
• Betweeness of Points
• Betweeness of Points
• Reflexive Property
• Addition Prop of
• Substitution

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Two Column Proofs

• Given AC BD
• Prove AB CD

.
.
.
.
A
D
B
C
Statement
Reason

• AC BD

• Given

2. AB BC AC
2. Betweeness of Points
3. Betweeness of Points
3. CD BC BD
4. BC BC
4. Reflexive Property
5. AC BC BD BC
5. Substitution
6. AB CD
6. Subtraction Prop of
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Two Column Proofs

• Given B is midpt of AC, C is midpt. of BD
• Prove AB ? CD

.
.
.
.
A
D
B
C
Statement
Reason

• B is midpt of AC, C is midpt. of BD

• Given

2. Def. of a midpoint
2. AB ? BC
3. Def. of a midpoint
3. BC ? CD
4. Transitive Property
4. AB ? CD
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Geometric Proofs

• Complete Worksheets on Line Segment Proofs