Section 11-1 – Basic Geometry Notions
Geometry: “geo” means earth and “metron” means measure
The fundamental building blocks of geometry are points, lines, and planes.
Linear Notions:
A line has no thickness, extends forever in two
directions and is determined by 2 points.
Points on the same line are called collinear.
A line segment is a subset of a line that contains 2 endpoints.
A ray is a subset of a line that contains one endpoint
and continues forever in one direction.
Planar Notions:
A plane has no thickness and extends indefinitely in two directions and is determined by 3 points.
Points and/or lines in the same plane are called coplanar.
Skew lines are lines that do not intersect and no plane contains them.
Intersecting lines are two coplanar lines with exactly one point in common.
Concurrent lines are lines that contain the same point (not necessarily all in the same plane).
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Properties of Points, Lines, and Planes:
1. There is exactly one line that contains any two distinct points.
2. If two points lie in a plane then the line containing the points lies in the plane.
3. If two distinct planes intersect, then their intersection is a line.
4. There is exactly one plane that contains any three distinct noncollinear points.
5. A line and a point not on the line determine a plane.
6. Two parallel lines determine a plane.
7. Two intersecting lines determine a plane.
A line and a plane can be related in one of 3 ways:
1. If they have no points in common, then the line and plane are parallel.
2. If two points of the line are in the plane, then the entire line is in the plane.
3. If only one point of the line is in the plane, then the line intersects the plane (cuts through).
Example: Sketch the following
a) Line AB is contained in Plane XYZ
b) Line AB and Plane XYZ have only one point in common.
c) Lines AB and CD are skew lines
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Example:
Given 10 points (no 3 of which are collinear) how many lines can be drawn through the 10 points?
•
Number of points
•
Number of Lines
2
3
•
4
•
•
5
6
•
•
•
•
7
8
9
10
•
•
•
•
•
Angles
When two rays share an endpoint an angle is formed. The angle contains sides and a vertex.
Adjacent angles share a common vertex and a common side and do not have overlapping interiors.
Angle measurement is usually done in degrees.
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Types of Angles:
(easy to illustrate by folding paper)
Right Angle (90°)
Straight Angle (180°)
Acute Angle (less than 90°)
Obtuse Angle (more than 90°)
Perpendicular lines occur when two lines intersect at a right angle.
A line perpendicular to a plane is a line that is perpendicular to every line in the plane through its
intersection with the plane.
Wall
Wall
Floor
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Example:
In the following figure,
PBR is 110°,
Find the measure of QBR.
QBS is 90°, and
PBS is 140°.
R
Q
S
P
B
Example: Angles A, B, and C are all adjacent angles that form a straight angle.
Find the measure of each angle if: A  53 B and B  12 C
Example: For the following figure:
B
A
C
D
a) Name a pair of parallel lines
F
G
b) Name a pair of skew lines
E
H
c) Name a pair of perpendicular planes
d) Are AB and FH parallel?
e) Do DH and plane ABE intersect?
f) Find the intersection of BH and plane ABC
g) Name two lines perpendicular to plane BC
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Section 11-2 – Polygons
Simple Curve: Does not cross itself, but it may have the same beginning and ending point
Closed Curve: Must start and stop at the same place
Simple Closed Curve: Does not cross itself and starts and stops at the same place.
Polygons: Simple and closed and have sides that are ONLY segments.
Convex: simple, closed, and has no indentations.
(the segment connecting any two points in the interior of the curve is completely contained in the
interior of the curve)
Concave: simple, closed, and has an indentation
Example: Classify the following
Shape
Simple
Closed
Polygon
Convex
Concave
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Polygons are classified according to the number of sides or vertices they have.
Each polygon has interior angles, exterior angles, and diagonals.
Congruency:
Two lines are congruent if they are the same length.
Two angles are congruent if they have the same measure.
AB  CD
ABC  XYZ
Polygons in which ALL interior angles are congruent and ALL sides are congruent are called
regular polygons.
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Definitions of Triangles and Quadrilaterals (see pg. 701-702)
Definition
A triangle with one right angle
Shape
Picture
A triangle in which all angles
are acute
A triangle with one obtuse
angle
A triangle with NO congruent
sides
A triangle with at least 2
congruent sides
A triangle with three congruent
sides
A quadrilateral with at least one
pair of parallel sides
A quadrilateral with two
adjacent sides congruent and
the other two sides also
congruent
A trapezoid with exactly one
pair of congruent sides
A quadrilateral in which each
pair of opposite sides is parallel
A parallelogram with a right
angle
A quadrilateral with all sides
congruent
A quadrilateral with four right
angles and four congruent sides
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The Heirarchy (see pg. 703)
Polygons
Quadrilaterals
Triangles
Trapezoid
Kite
Scalene
Isosceles
Parallelogram
Isosceles
Trapezoid
Equilateral
Rectangle
Rhombus
Square
Example: True or False?
____________ An equilateral triangle is isosceles
____________ A square is a rectangle
____________ A rectangle is a square
____________ A square is a regular quadrilateral
____________ A square is a rhombus with a right angle
____________ If a kite has a right angle, then it must be a square.
____________ The base angles in an isosceles triangle are congruent.
____________ A parallelogram is a square.
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Example: Relationship between the number of sides and the number of diagonals in a polygon.
Complete the following chart:
Number of Sides
Number of Diagonals
3
4
5
6
7
8
9
N
Example: True or False?
The number of diagonals in a figure is greater than the number of sides in that figure.
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Section 11-3 – More About Angles
Two angles whose sum is 180° are called
______________________________________.
Two angles whose sum is 90° are called
______________________________________.
Vertical Angles are always congruent
Transversal: A line that intersects a pair of lines in a plane
Interior Angles:
Exterior Angles:
Alternate Interior Angles:
Alternate Exterior Angles:
Corresponding Angles:
Theorem: If any two distinct coplanar lines are cut by a transversal, then corresponding angles are
congruent, alternate interior angles are congruent, and alternate exterior angles are congruent
IF AND ONLY IF the lines are parallel.
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Example: In the following figure n || m. Explain why angles 1 and 2 are supplementary.
N
1
2
M
Example: Are the following lines parallel?
n
40°
30°
70°
m
The sum of the measures of the interior angles of a triangle is _____________.
Proof #1 (inductive reasoning) Tearing Paper
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Proof #2 (deductive reasoning)
The Relationship between the sum of the Interior Angles and the sum of the Exterior Angles:
Example: Complete the following chart:
Number of Sides in
the polygon
Sum of the Measures of
the Interior Angles
Sum of the Measures of
the Exterior Angles
3
4
5
6
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Theorem:
The sum of the measures of the exterior angles of a convex polygon is _______________.
The sum of the measures of the interior angles of any convex polygon with n sides is:
__________________________
or
____________________________
The measure of a SINGLE interior angle of a regular n-gon is:
___________________________ or ____________________________
Example: Find the measure of each interior angle of a regular decagon.
Example: If the sum of the measures of the interior angles in a regular polygon is 1980°, then how
many sides does the polygon have?
Example: For the following parallelogram, if angle A is 40°, then find the measures of all of the
other angles.
C
A
D
B
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Section 11-4 – Geometry in Three Dimensions
Simple Closed Surfaces have exactly one interior, no holes, and are hollow. (ping pong ball)
Sphere: The set of all points at a given distance (radius) from a given point (center).
Solid: The set of all points on a simple closed surface along with all interior points. (ball bearing)
Polyhedron: A simple closed surface made up of polygonal regions, or faces.
It contains vertices and edges.
See pg. 727 for examples
Prism: A polyhedron in which two congruent faces (bases) lie in parallel planes and the other faces
are bounded by parallelograms. A prism is usually named after its base(s). Prisms can be right or
oblique.
Pyramid: A polyhedron determined by a polygon and a point not in the plane of the polygon. It
contains triangular regions. The point is called the apex.
A Convex Polyhedron occurs if the segment connecting any two points in the interior of the
polyhedron is itself in the interior
A Regular Polyhedron is a convex polyhedron whose faces are congruent regular polygonal
regions such that the number of edges that meet at each vertex is the same for all vertices.
See pg. 729
Cube, Tetrahedron, Octahedron, Dodecahedron, Icosahedron
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Example: Given the tetrahedron below, name the:
A
a) Vertices
B
C
b) Edges
D
c) Faces
Cylinders and Cones are not polyhedrons.
Cylinders have bases that are simple closed curves and parallel sides.
Cones have bases that are simple closed curves and meet at a vertex.
Cylinders and cones can be right or oblique.
Example: Build the following then complete the following chart:
Name
# of Vertices
# of Faces
# of Edges
Tetrahedron
Cube
Square Pyramid
Octahedron
Pentagonal Pyramid
Hexagonal Prism
Dodecahedron
Icosahedron
The relationship between the number of vertices, faces, and edges is known as:
Euler’s formula: _________________________
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Example: Which polyhedral can be formed from the following nets?
a)
b)
c)
d)
e)
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