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Copy of Geometry chapter 1-3
Transcript of Copy of Geometry chapter 1-3
Points, Lines, and Planes
Collinear and Coplanar points
Segments / Rays / Opposite Rays
Name the lines that are considered parallel, perpendicular and intersecting?
What’s formed by intersecton of two planes?
Example: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
A rule that can be proved is a called a theorem, as you will see later.
Postulate 1 - Ruler Postulate
1. You can pair each point on a line with a real number. The number that corresponds to the point is called the point's coordinate.
2. To find the distance between two points: subtract their
coordinates, and take the absolute value (make the answer a
Postulate 2 - Segment Addition Postulate
If Q is between P and R, then PQ + QR = PR.
If PQ + QR = PR, then Q is between P and R.
Line segments that have the same length are called congruent segments.
In the diagram below, what can we say about these segments?
What can we say about "lengths that are equal" and "segments that are congruent"?
AM = 3
MB = 3, so
AM = MB
These are all numbers which
can be added, subtracted,
multiplied, divided etc.
AM MB - "is congruent to"
these are geometric objects
existing in space; they cannot
be added, subtracted, multiplied,
Segments and Congruence
Midpoint: On a number line, the number halfway
between x1 and x2 is (x1+x2)/2. How do we calculate it
on a plane?
Definition of Midpoint
* Midpoint of a segment is the point that divides the segment into two congruent segments.
Definition of segment bisector
* a segment bisector is a point, ray, line, segment or plane that intersects the segment at its midpoint.
Segment Bisector/ Perpendicular Bisector:
Pythagorean Theorem & Distance Formula
What is the distance between the points (5, 6)
and (– 12, 40) ?
An angle consists of two different rays with the same end point.
The rays are the sides of the angle. The endpoint is the vertex of
of the angle.
The angles with sides ray AB and ray AC can be named angle BAC,
CAB, or A. Point A is the vertex of the angle.
Name three angles above.
Postulate 3 - Protractor Postulate
Consider ray OB and a point A not on ray OB. The ray that forms OA can be matched one to one with real numbers from 0 to 180.
The measure of angle AOB is equal to the absolute value of the difference between the real numbers ray OA and ray OB.
Measuring and Classifying Angles
Angles can be classified as acute, right, obtuse and straight.
When are angles congruent?
Postulate 4 - Angle Addition Postulate
WORDS - If P is in the interior of the angle RST, then the measure of angle RST is equal to the sum of the measures of angle RSP and angle PST.
SYMBOLS - If P is in the interior of angle RST, then
m RST = m RSP + m PST.
If you were at the South Pole: What longitude degree would use to get home to Brazil? What about if you lived in Australia?
Constructing congruent segments and a perpendicular bisectors.
Constructing congruent angles and an angle bisector.
Ready to do something fun???
How did that little brain teaser work out for you???
Complementary angles are two angles who's sum is 90 degrees.
Supplementary angles are two angles who's sum is 180 degrees.
Adjacent angles are two angles that share a common vertex and side, but have no common interior points.
Two adjacent angles are a linear pair if their common sides are opposite rays.
Vertical angles are two nonadjacent angles formed by two intersecting lines.
Wait...how about a brain teaser today??? Up for it???
What is a polygon?
A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is nonconvex or concave.
A polygon is a closed figure with N sides and the following properties. Where N represents the numbers of sides.
It is formed by three or more line segments called sides.
Each side intersects exactly two sides, one at each endpoint.
Each endpoint of a side is a vertex of the polygon. The plural of vertex is vertices. A polygon can be named by listing the vertices in consecutive order. For example, ACEFDB and EFDBAC are both the correct names for the polygon below.
What is an equilateral polygon?
What about an equiangular polygon?
How much would you pay for this chaise (lounge chair)?
A perimeter is a path that surrounds an area. The perimeter of a circular area is called circumference.
Area is the amount of surface space that a flat
Find Perimeter, Circumference, and Area
If I were right here, at the exact midpoint of Maracana Stadium, how many meters would I have to walk to get off the field walking towards the goal? How many meters walking towards the sidelines?
Consider the pattern suggested by the following figures:
# of points connected: 2 3 4 5 6
# of regions formed: 2 4 8 16 ?
Inductive reasoning is when you find a pattern in specific cases and then write a conjecture for the general case.
A conjecture is an unproven statement that is based observations.
To show conjecture is true, you must show that it is true in all cases. You can show conjecture is false, however, by simply finding one counterexample.
A counterexample is a specific case for which conjecture is false.
Only 31 regions!!!
Magic Number Trick
I bet I can guess the number you are thinking if you do the following:
* Think of a number.
* Add 7 to it.
* Subtract 2.
* Subtract your original number.
* Multiply by 4.
* Subtract 2.
Prediction: Your result will be 18
You want to know why?
This problem gives the illusion of guessing an unknown number, but the unknown has been cancelled mathematically.
( X + 7 - 2 - X ) × 4 - 2 = 18
A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion.
Conditional statements are written in if-then form. The "if" part contains the hypothesis and the "then" part contains the conclusion.
Negation, Converse, Inverse, Contrapositive and Biconditional
Statement: A right angle measures 90º. (true)
Negation: A right angle does not measure 90º. (false)
Conditional: If an angle measures 80º, then it is acute. (true) (If p, then q or p q )
Converse: If an angle is acute, then it measures 80º. (false) (If q, then p or
q p )
Inverse: If an angle does not measure 80º, then it is not acute. (false) (if ~p, then ~q)
Contrapositive: If an angle is not acute, then it does not measure 80º. (true) (if ~q, then ~p)
Truth value: conditional/contrapositive, converse/inverse
Conditional: If an angle is straight, then it measures 180º.
Converse: If an angle measures 180º, then it is straight.
Inverse: If an angle is not straight, then it does not measure 180º.
Contrapositive: If an angle does not measure 180º, then it is not straight.
Biconditional: An angle is straight if and only if it measures 180º.
Deductive Reasoning: Uses facts, definitions, accepted properties and the laws of logic to form a logical argument.
If <A and <B are complementary and m<A is 35º, then m<B is 55º.
m<A +m<B= 90º -Definition of complementary Angles
35º +m<B=90º -Substitution Property
m<B=55º -Subtraction Property
Apply Deductive Reasoning
Inductive reasoning works moving from specific observations to broader generalizations and theories. We sometimes call this a "bottom up" approach. In inductive reasoning, we begin with specific observations and measures, detect patterns, formulate hypotheses, explore them, and finally end up developing some general conclusions or theories.
Deductive reasoning works the other way from the more general to the more specific. We call it a "top-down" approach. We begin with thinking up a theory about our topic of interest. We then narrow that down into more specific hypotheses that we can test.
Logic is the anatomy of thought.
– Albert Einstein
Chapter 2.5 - 2.7
Reason Using Properties from Algebra
Addition Property of Equality If a = b, then a + c = b + c.
Subtraction Property of Equality If a = b, then a – c = b -– c.
Multiplication Property of Equality If a = b, then a * c = b * c.
Division Property of Equality If a = b, then a/c = b/c.
Substitution Property of Equality
Reflexive Property of Equality For any real number a, a = a.
For any segment AB, AB = AB.
For any angle A, m<A = m<A.
Symmetric Property of Addition For any real numbers a and b, if a = b, then b = a.
For any segments AB and CD, if AB = CD, then CD = AB.
For any angles A and B,if m<A = m<B,then the m<B = m<A.
Transitive Property of Addition
Section 2.6 Prove Statements about
Segment and Angles
A proof is a logical arguement that shows a statement is true.
Two column proofs have numbered statements and corresponding reasons that show arguement in logical order.
The reasons used in a proof can be definiitions, properties, postulates and theorems.
A theorem is a statement that can be proven.
Prove Angle Pair Relationships
Theorm 2.3 Right Angles Congruence
All right angles are congruent.
Theorem 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
Theorem 2.5 Congruent Complements Theorem
If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
Theorem 2.6 Vertical Angles Congruence Theorem
Vertical angles are congruent.
Postulate 12 Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
Identify Pairs of Lines and Angles
Two lines that do not intersect are either parallel lines or skew lines.
Parallel lines are two lines that do not intersect and are coplanar.
Skew lines are two lines that do not intersect and are not coplanar.
Postulate 13 - Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
Postulate 14 - Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.
A transversal is a line tht intersects two or
more coplanar lines at different points.
Can you name the following from the diagram above?
Alternate interior angles
Alternate exterior angles
Consecutive interior angles
(Same side interior angles)
Brazilian National Congress
Catedral de Brasil
US National Cathederal
US Capital Building
Use Parallel Lines and Transversals
Postulate 15 - Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Theorem 3.1 - Alternate Interior AnglesTheorem
Theorem 3.2 - Alternate Exterior Angles Theorem
Theorem 3.3 - Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
If two parallel lines are cut by a transversal, then the pairs of aconsecutive interior angles are supplementary.
Source - http://www.regentsprep.org/regents/math/geometry/GP8/Lparallel.htm
Prove Lines are Parallel
Postulate 16 - Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
Theorem 3.4 Alternate Interior Angles Converse
Theorem 3.5 Alternate Exterior Angles Converse
Theorem 3.6 Consecutive Interior Angles Converse
If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
If two lines are cut by a transversal so the consecutive interior angles are supplementrary, then the lines are parallel.
Is it difficult to prove something? What about proving Geometry?
Does that make sense?
Can you use this to prove this?
Proofs are fun, right?
Find and Use Slopes of Lines
The slope of a nonvertical line is the ratio of the vertical
change (rise) to the horizontal change (run) between any two points on a line.
If a line in the coordinate plane passes through points (x1, y1) and (x2, y2) then the slope m is:
m = rise/run = change in y / change in x = y2-y1 / x2-x1
Slope of Lines in the Coordinate Plane
Positive slope: rises from left to right.
Negative slope: falls from left to right.
Zero slope (slope of 0):
U s v
n l e
d o r
e p t
f e i
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope.
Any two veritcal lines are parallel.
Postulate 18 - Slopes of Perpendicular Lines
In a coordinate plane, two nonveritical lines are perpendicular if and only if the product of their slopes is -1.
Horizontal lines are perpendicular to veritcal lines.
Slopes like these...
Maybe like these...
Write and Graph Equations of Lines
General form of a linear equation
y = mx + b,
where m is the slope and b is the y-intercept
Find the slope. Choose two points on the graph of the line.
m = _________
Find the y-int. The line intersects the y-axis at what point?
b = _________
Write the equation.
y = mx + b
Can you write the equation of the line that is parallel to the line AB to the left?
How about a perpendicular line to that line with the same y-intercept?
Another form of a linear equation is the standard form, which is written as:
Ax + By = C
To find the x-intercept,
let y = 0
To find the y-intercept,
let x = 0.
x = ___ y = ___
The line intersects the axis at (4,0) and (0,3). Confirm we have the correct intersections.
Can you do the same for the following equation?
2x - 3y = 6
How about this one?
x = -3
y = 4
The equation y = 50x + 125 models the total cost of joining a climbing club. What are the meanings of the slope and the y-intercept?
Y = 50X + 125
Prove Theorems About Perpendicular Lines
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
If APQ BPQ,
then QP AB.
If two lines are perpendicular, then they intersect to form four right angles.
If line 1 line 2, then <3, <4, <5 and <6 are right angles.
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
If BA BC, then <1 and <2 are complementary.
If a transversal is perpendicular to one of two parallal lines, then it is perpendicular to the other.
If k l and t k, then t l.
Lines Perpendicular to a Transversal
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
If M P and N P,
then M N.
Postulate/Axiom: Accepted without proof
Identify Points, Lines and Planes
Use Segments and Congruence
Use Midpoint and Distance Formulas
Measure and Classify Angles
m < RST
m < RSP
m < PST
Describe Angle Pair Relationships
Find Perimter, Circumference, and Area
Use Inductive Reasoning
A student makes the following conjecture about the sum of two numbers.
The sum of two numbers is always greater than the larger number.
Let's try an Example.
-2 + -3 = -5,
so -5 is greater then -2, wrong - because a counterexample exists, the conjecture is false.
-7, -21, -63, -189, ...
(What's the next three numbers in this pattern?)
Your favorite sports team has won 6 games on a row. Do you think they will win or lose the 7th game? If you answered "win", you used inductive reasoning.
Analyzing Conditional Statements
When n=9, n =81.
If n=9, then n =81
All birds have feathers.
If an animal is a bird, then it has feathers.
Opposite to the original statement.
Exchange hypothesis and conclusion.
Negate the conditional statement.
Negate the Converse Statement.
When the conditional and converse statements are true, we can write the biconditional statement; ´if and only if.` (p q)
Hint...see page 149
(See page 80)
Laws of Logic
Law of Detachment
If the hypothesis of a true conditional statement is true, then the conclusion is also true.
Law of Syllogism
If hypothesis p, then conclusion q.
If hypothesis q, then conclusion r.
If hypothesis p, then conclusion r.
If these statements are true,
then this statement is true.
Is this how you calculate?
If a = b, then a can substituted
for b in any equation or expression.
For any real numbers a, b and c, if a = b and b = c, then a = c.
For any segments AB, CD and EF, if AB = CD, then AB = EF.
For any angles A,B and C, if the m<A = m<B and the m<B = m <C,then m<A = m<C.
Theorm 2.1 Congruence of Segments
(Theorem 2.2 Congruence of Angles)
Reflexive - for any segment AB, AB =AB
(same true for any <A = <A)
Symmetric - if segment AB = segment CD, then segment CD = seg AB.
(same is true for <A = <B, then <B = <A)
Transitive Property of Congruence: If seg A =seg B and seg B =seg C, then seg A = seg C.
(same is true for <A = <B and <C = <B, then <A = <C)
Postulate 17 - Slopes of Parallel Lines