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Geometry, Chapter 2
Section 2.1: Axiomatic Systems; definitions and notation for geometric figures; types of angles
1.
An axiomatic system (also called a deductive reasoning system) consists of undefined terms, definitions,
axioms or postulates, and theorems.
a. Undefined terms – can be described but cannot be given precise definitions. The properties of undefined
terms are given by the postulates or axioms of the system.
Set – a collection of objects
Point – determines a position but that has no dimension (length, width, or height).
Line – set of points in a one-dimensional figure with no thickness that extends in opposite directions
without ending. It is usually straight, but that depends on the postulates.
Plane – set of points in a surface having two dimensions (length and width) that extend in all
directions without ending.
b. Definitions – statements that give precise meaning using undefined terms and other definitions. Here are
two examples:
Space – the set of all points.
Geometric Figure – any set of points, lines or planes in space.
c. Postulates or Axioms – statements about undefined terms and definitions that are accepted as true
without verification or proof.
d. Theorems – a statement that we can prove using definitions, postulates, previously proved theorems and
the rules of deduction and logic.
2.
Postulates from the Textbook
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
Every line contains at least two distinct points.
Two points are contained in one and only one line.
If two points are in a plane, then the line containing these points is also in the plane.
Given any three distinct points in space not on the same line (noncollinear), there is exactly one plane that
passes through them.
No plane contains all points in space.
(The Ruler Postulate) There is a one-to-one correspondence between the set of all points on a line and the set
of all real numbers. (This postulate implies every line contains an infinite number of points and that every line
is infinitely long.)
3.
Collinear points: two or more points that lie on the same line.
Coplanar points: three or more points that lie on the same plane.
4.
Line Segment: A portion of a line that consists of two points, A and B, and all points that are between them.
A and B are called the endpoints.
Notation: line through points A and B: AB ,
5.
line segment from A to B: AB
Congruent Line Segments: Two lie segments with the same length. The symbol for congruency is: ≅ .
Notation: “length of segment AB ” is AB (when it is written with the bar above the endpoints it
refers to the collection of points in the segment, without the bar it refers to the length of the
segment. So “ AB ≅ CD ” (the segments are congruent) means the same thing as “AB = CD”
(their lengths are equal).
6.
Midpoint: A point that divides a line segment into two congruent segments.
7.
Ray: a portion of a line that begins at some point, A, and continues forever in one direction. A is called the
endpoint. Notation: The ray from point A through point B: AB
Geometry, Chapter 2
8.
Angle: A figure formed by two rays with a common endpoint. The common endpoint is called the vertex of
the angle and the rays are called the sides.
9.
Congruent Angles: Two angles with the same measure.
10.
Types of Angles
Acute: An angle whose measure is between 0° and 90°.
Right: An angle whose measure is 90°.
Obtuse: An angle whose measure is between 90° and 180°.
Straight: An angle whose measure is 180°.
Reflex: An angle whose measure is between 180° and 360°.
11.
Angle Bisector: A ray that divides an angle into two congruent angles.
12.
Adjacent Angles: Two angles that have a common endpoint and a common side lying between them.
13.
Perpendicular Lines: Two lines that form a right angle.
(An alternate definition for perpendicular lines is “two lines that intersect and form congruent adjacent
angles.” Sketch two intersecting lines until this definition makes sense to you.)
14.
Parallel lines: Two lines in a plane that do not intersect.
15.
Complementary Angles: Two angles whose sum is 90°.
Supplementary Angles: Two angles whose sum is 180°.
Section 2.2: types of triangles and quadrilaterals; definitions associated with polygons and circles
1.
Types of Triangles:
Right Triangle: A triangle with one right angle.
Hypotenuse: The side opposite the right angle in a right triangle.
Legs: The perpendicular sides of a right triangle.
Acute Triangle: A triangle in which all three angles are acute or less than 90°.
Obtuse Triangle: A triangle in which one angle is greater than 90°.
Scalene Triangle: A triangle in which all three sides have different lengths.
Equilateral Triangle: A triangle in which all three sides are the same length (congruent).
Equiangular: A triangle in which all three angles have the same measure (congruent).
Isosceles Triangle: A triangle with at least two congruent sides.
Base Angles: The angles opposite the congruent sides of an isosceles triangle.
Vertex Angle: The angle other than the base angles of an isosceles triangle. The side opposite the
vertex angle is called the base.
2.
Simple Closed Curve: A figure that lies in the plane and can be traced so that its starting and ending points
are the same and no part of the curve is crossed or retraced.
3.
Polygon: A simple closed curve composed of line segments. Some common polygons and number of sides:
Triangle: 3
Hexagon: 6
Nonagon: 9
Quadrilateral: 4
Heptagon: 7
Decagon: 10
Pentagon: 5
Octagon: 8
Dodecagon: 12
Geometry, Chapter 2
4.
5.
Circle: a simple closed curve consisting of the set of all points equidistant from a
given point, the center.
Radius: a line segment that joins the center with a point on the circle
Diameter: a segment that contains the center and has its endpoints on the circle
arc
B
chord
central
angle
radius O
diameter
A
C
Quadrilateral: A polygon with four sides.
semicircle
Square: A quadrilateral with all sides congruent; all angles are right angles.
Rectangle: A quadrilateral in which all angles are right angles.
Rhombus: A quadrilateral with all sides congruent.
Parallelogram: A quadrilateral with parallel opposite sides.
Trapezoid: A quadrilateral with exactly one pair of sides parallel. The parallel sides called bases and the
other sides are called legs.
Isosceles Trapezoid: A trapezoid in which the nonparallel sides are congruent.
Kite: A quadrilateral with exactly two distinct pairs of congruent adjacent sides. The angles between the two
congruent sides are called vertex angles.
Section 2.3: terminology and formulas for vertex angles for polygons
1.
Angle Sum in a Triangle The sum of the measures of the angles of a triangle is 180˚. (We will prove this
in Chapter 5.)
2.
Vertex of a Polygon – the point where two sides intersect.
Vertex Angle or Interior Angle of a Polygon – the angle formed by two adjacent sides.
3.
Diagonal of a Polygon – a segment connecting two non-adjacent vertices.
4.
Angle Sum in a Polygon: The sum of the measures of the vertex angles in any polygon with n sides is
(n − 2) ⋅180O .
5.
Regular Polygon: All sides are congruent and all vertex angles are congruent. Since all vertex angles have
equal measures we can modify the formula above as follows:
(n − 2) ⋅180O
The measure of each vertex angle in a regular polygon is
n
6.
Center of a Regular Polygon - i. A point that is equidistant from all angles in the
regular polygon and equidistant from all sides.
ii. The center of the circle circumscribed about the regular polygon.
7.
Radius of a Regular Polygon – A line segment drawn from the
center of the polygon to one of its vertices.
8.
Central Angle of a Regular Polygon - The angle formed by radii
drawn to two consecutive vertices.
9.
Exterior Angle of a Polygon – The smaller angle that is adjacent to
an interior angle, formed by extending a side of the polygon.
B
A
C
O
F
c
D
E
S
A
exterior angle
B
interior angle
F
C
E
D
Geometry, Chapter 2
Section 2.4: 3-Dimensional Shapes
1. A polyhedron is a 3-dimensional shape composed of polygons (faces).
2. Euler’s formula: F + V = E + 2 where F = # faces, V = # vertices, E = # edges (sides of polygons)
Ex: for a cube or rectangular box: F = 6, V = 8, E = 12 and 6 + 8 = 12 + 2
3. Prism: Two congruent polygons form opposite, parallel bases connected by lateral faces which are
quadrilaterals. The height, h, is the distance between the bases.
h
h
Right Prism
Lateral faces are rectangles
Oblique Prism
Lateral faces are parallelograms
4. Pyramid: One polygon base, with triangular lateral faces meeting at the apex. The height, h, is the
perpendicular distance from the base to the apex.
l
h
h
Right Regular Pyramid
1. Base is a regular polygon
2. Lateral faces are isosceles triangles
3. Slant height, l, is the height of the lateral faces
Oblique Regular Pyramid
1. Base is a regular polygon
2. Lateral faces are not isosceles triangles
3. No slant height, lateral faces not congruent
5. Cylinders and Cones:
Right Circular Cylinder
Oblique Circular Cylinder
Right Circular Cone
l
h
h
h
Oblique Circular Cone