Geometry A IHS OL Unit 1 & 2

point

-location in space, precise location or place on plane

line

set of points arranged along straight path
-no thickness
-goes in both directions w/o end
-infinitely thin and long
-1 dimensional geographic object
-symbol: two letters with arrow above, or line n

Ray

Endpoint on one end, but goes in 1 direction forever
-part of a line
-opposite rays start at same point but go in opposite direction

angle

two rays that started at the same point
-vertex = initial point
-sides are made up of rays

• <A or <BAC

Line segment

point of a line with 2 endpoints
-finite
-can be written in any order

plane

flat surface that extends infinitely
-no length/width, no curve
-label 3 noncolinear points
-order does not matter

Colinear

if they are on the same line, opposite is noncolinear

Coplanar

Same plane, non-coplanar is opposite

Conjecture

Assertion that is most likely true but has not been proven yet
-2,4,6

Three proofs

Definitions
Postulates
Axioms

Equilateral triangle

three sides that are equal in measure

Right angles

When a line intersects another to form adjacent angles with equal measure
-lines are perpendicular

axioms/postulates

-interchangeable
-accepted statement of fact that cannot be disproved

4 postulates

1. A straight line may be drawn from any given point to any other point

2. A straight line may be extended to any finite length

3. A circle may be described with any given point as its center and any distance as its radius

4. All right angles are congruent
   -Extra: if it is not labeled as a right angle, you cannot claim it is a right angle.

Betweeness Postulate

If B is between A and C, then AB + BC = AC. If AB + BC = AC then B is between A and C

Octagon

8 sides

Rectangle

A parallelogram with four right angles and four sides

Square

Four sides of equal length

Polygon

Closed figure formed by three or more line segments called sides that are connected by endpoints called vertices

Adjacent sides

Sides that appear next to each other

Vertex

Endpoints of a side. Put in order with letter

interior angles

formed at each vertex of polygons

exterior angles

formed between any side of polygon and extension of adjacent side

regular polygons

congruent sides and interior angles that have equal measures (equilateral and equiangular)

irregular polygons

ones that have sides with different lengths and interior angles of different measures

concave polygon

one or more interior angles greater than 180, looks like pushed inward

convex polygons

ones that have all interior angles measuring less than 180, vertices point outwards from interior of polygon

simple polygons

ones that have sides that do not intersect or cross with each other connected by vertices

complex polygons

ones that have sides that intersect or cross to form smaller figures

triangle

A polygon with three sides.

quadrilateral

4 sides connected by endpoints

pentagon

5 sided polygon

hexagon

6 sided polygon

heptagon

7 sided polygon

octagon

a polygon with 8 sides

nonagon

9 side polygon

decagon

10 sides polygon

n-gon

a polygon with n sides

parallelogram

opposite sides are parallel (coplanar, same distance apart)

rhombus

parallelogram w/ four sides of equal length

rectangle

parallelogram w/ two pairs of opposite sides that are equal in length and has four right angles

square

parallelogram four sides of equal length and four right angles

trapezoid

A quadrilateral with exactly one pair of opposite parallel sides

kite

a quadrilateral with two pairs of adjacent sides that have the same length

circle

set of points on a plane that are given a distance from a given point called the center
-two circles with the same radius = same
-if a radius is larger than another, it is the larger circle and vise versa

ellipse

closed curve like an oval
-result of keeping the sum of two lengths, AP and PB constant while changing the individual lengths of AP and PB (generator lines)

foci

two points that form the shape of an ellipse

minor axis and major axis

-minor, short diameter
-major, long diameter
-connect at center point

semi-major and semi-minor

-semi major: 1/2 of major axis, longest radius
-semi minor, 1/2 of minor axis, shortest radius

Congruent

Identifal in size and shape to another object

Similar

Identical shape with corresponding angles and proportional sides
-different size
-congruent always is similar
-similar not always congruent, angles may have same measurements but diff side lengths
-scaled copies

How to use a protractor

-larger than 90 = obtuse, use numbers 90+
-less than 90 = acute, use numbers -90

How to see if figures have proportional sides

Short side of small figure / short side of large figure = long side of small figure / long side of big figure
-if equal = proportional
-2/3 =6/9=2:1

traits of a parallelogram

-both sets of opposite sides are parallel to each other
-both sets of opposite angles/sides are congruent
-supplementary angles (add up to 180) are formed by consecutive interior angles

perimeter

around the edge
-p=2L + 2W

area

The number of square units required to cover a surface
-a=lxw or a=bxh = units squared
-for a parallelogram, has to be perpendicular to base. height has to form right angle
-area for triangle: 1/2bh or bh/2
-area of circle: pieR^2

circumference

distance around circle
-c=pi times diameter
-pi= c/d
-area of a circle: A=pier^2

pre image

original, pre image turns into image, makes a transformation

reflections

mirrored objects over lines of reflection
-y=x: (x,y) --> (y,x)
-x axis: -x, y
-y axis: x, -y

rotations

turned objects clockwise/counterclockwise
-clockwise: left to right circularly
-cc: right to left circularly
Counterclockwise rules:
-90: -y,x, 180: -x,-y (same for clockwise), 270: y, -x
Clockwise rules:
-270: -y, x
-90: y, -x

translation

move a specific distance

constructions

drawing of geometric figures such as lines and circles using only a compass and straightedges

bisect

divide angles or segments into halves
-Refer to L8 U1 to see how to bisect things.

Refer to Unit 1 Lesson 9 to see how to construct multiple things

Truth table

Only can be true or false
-Helps you draw correct conclusions

Symbols and their meanings

~p = not p, opposite of p
-^ and
-^ (flipped vertically) or

• —> if then or implied

Conjunction and disjunction rules

-conjunctions are true when both statements are true
-disjunctions are only false when both statements are false

Refer to Unit 1 Lesson 10 to see conjunctions, disjunctions, conditional, logic puzzles, and truth tables

angle

shape formed by two rays diverging from two points known as the vertex

initial and terminal side

intial-base of the angle
terminal-the one that inclines

positive and negative angle

-pos: terminal side inclines counterclockwise from initial side
-neg: terminal side inclines clockwise from initial side

How to draw an angle

1. Draw a line segment

2. Find a degree on the protractor and mark a spot where it is

3. Draw your line connecting from the vertex to the marked spot

acute

less than 90 degrees

obtuse

more than 90, less than 180 degrees

straight angle

flat, 180 degrees

right angle

90 degree angle

reflex angle

more than 180, less than 360

Adjacent angles

-common vertex and side
-can also be nonadjacent

Interior & exterior angles

-inside and outside a shape/lines

Complementary angles

Pair of angles that equal 90 degrees
-can be nonadjacent or adjacent

supplementary angles

two measures/angles that add up to 180 degrees
-can be nonadjacent or adjacent

Transversal

a line that intersects two or more lines

Consecutive interior and exterior angles

-interior: inside lines on the same side of transversal
-exterior: outside lines on the same side of transversal
-supplementary

Alternate interior and exterior angles

-interior: inside, opposite side
-exterior: outside, opposite side
-congruent to each other

Corresponding angles

Angles in the same place on different lines
-same position
-congruent

verticle angles

are opposite angles of two intersecting lines
-form an X like shape
-congruent

Property

Mathematical statement or equation that describes characteristics of two quantities

Reflexive Property

An angle is congruent to itself

Symmetric property

Angles are congruent even if they are different size, use measurements to see congruency

Transitive property

If a=b and b=c, then a=c

Postulate

-statement not proven, but accepted with logic and reasoning
—> angle addition: if ray BD divides angle ABC then < ABD and < DBC will measure ABC

Theorem

Conjecture that has been proven true and therefore can be used to prove additional conjectures

Steps of proofs

1. State conjecture to be proved

2. List all given info

3. Draw a diagram, update it with additional info

4. Deduce additional statements "Where can I go from here"

5. Stop when conjecture is true

Parallel lines

Lines that extend in same direction and never touch
-coplanar
-equidistant: same distance from each other as you measure along lines

Intersecting lines

Line in same plane but intersect each other, not equidistant

Perpendicular lines

Lines that intersect at right angles

Skew

Lines that do not intersect and are not coplanar, not parallel

Refer to notes to see how to construct parallel lines, perpendicular lines, and a perpendicular bisector

perpendicular bisector

bisector that intersects a segment at a right angle into two equal haves
-intersects at a midpoint

congruent

exactly same size and shape
-use equal sign for angle MEASURES m < CDA = m < CDB
-for angles/segments themselves, use congruent symbol <CDA --~ <CDB