Download Introduction to Geometry

Geometry A Overview
General Formulas
Given P=(x.yi) and Q (x2,y2), distance is d
•
1)istance Formula:
•
Midpoint Formula: Given P=(xi,yi) and
Q
•
Slope Formula: Given P(xj,yi) and
(x2,y2), slope is m
2 +(y
1
1 —x,)
/(x
=
Q
±X
2
, midpoint is
,y
2
(x
)
‘
) ±
2
=
1
X
—
Symbols
Meaning
Not equal
Congruent
Greater than or equal to
Less than or equal to
Not
And
Or
ymboI
E
A
v
Meaning
Triangle
Angle
Perpendicular
Union
Intersection
Implies
If and only if
A
X
I
U
fl
Symbol
..
•:
3
\/
Meaning
Therefore
Since
There exists
Such that
For all or for every
Is an element of
Is not an element of
Introduction to Geometry
Vocabulary
• Angle (I): lbrmed by 2 rays that have same endpoint
• Bisect: to divide segment or angle into 2 congruent parts
• Bisector: point or ray that bisects segment or angle
• Collinear: points on same line
• Congruent (): same measure
• Line (—>): made up of points and is straight (symbol: arrowhead at each end)
• Line Segment/Segment (): has 2 endpoints, can be measured. (named after 2 endpoints)
• Midpoint: point that bisects a segment. Must be collinear
• Noncollinear: points that don’t lie on same line
>): begins at an endpoint & then extends infinitely far in only 1 direction.
• Ray(
•
•
•
Tick Marks: indicate congruency
Trisect: to divide segment or angle into 3 congruent parts
Trisectors: points or rays that trisect segment or angle
Algebraic Phrases
Engsh Word Algebraic Translation
90 —x
Complement
DitYerence
=
Equal
+,>
than
Greater
+
Increased by
Less than
-
—
English Word
Number
Opposite of a number
Product
Sum
Supplement
Algebraic Translation
N
-N
*
+
180— x
-,
Example: Angle Measures
I. Find the measures of 2 supplementary angels if the measure of I angle is 6 less than 5 times the measure of
the other angle.
180—x = 5x -6; anglesl49, 31
RevB
GeQrnetry A Overview
Angle Relationships
Vocabulary
• Acute Angle: between 0° and 90°
• Adjacent Angles: 2 / that lie in same plane, have common vertex & side but no common interior point
• Complementary Angles: 2 angles whose measures have sum of 90°
• Linear Pair: pair of adjacent angels whose non-common sides are opposite rays.
• Obtuse Angle: between 90° and 180°
• Perpendicular (I): lines that form 90° angles. Intersect to form congruent adjacent angles; rt angle symbol
indicates lines are perpendicular
• Right Angle: 90°
• Straight Angle: 180°
• Supplementary Angles: 2 angles whose measures have sum of 180°
• Vertical Angles: 2 nonadjacent angles formed by 2 intersecting lines.
Example: Angle Relationships
Given the diagram to the right, identify the following:
1. Adjacent angles: / AED & / AEB, / CED & / CEB
2. Vertical Angles: / DEA & r CEB, / AEB & / CED
3. Linear Pair: / DEA & / BEA,
r DEC &
/ BEC
4. Given the diagram to the right, identify all angle relationships.
• / I and / 5 are exterior angles
• / I sup /2 / I + /2 =180, / I & r 2 are linear pair
• r 4 sup /5 /4 + /5 =180. /4 & /5 are linear pair
• /2±/3+Z4=180
• /1=Z3±/4,/l>/3,Z1>/4
• /5rr Z2+ /3, /5> /2, /5>3
/3
.
//
.
<
1/2
4
5
CM
Reasoning & Proofs
Vocabulary
•
•
•
•
•
•
•
•
•
•
•
•
Compound Statement: 2 or more statements joined together
Conclusion: phrase immediately following then
Conditional Statement/implication: if-then statement
Conjecture: educated guess based on known info
Conjunction (A): compound statement formed by joining 2 or more statements with word and; true when
both are true
if-q then —p
Contrapositive: if p then q
Converse: statement associated with if p then q having form if q then p
I)isjunction (v): compound statement formed by joining 2 or more statements with word or; true when at
least 1 statement is true
Hypothesis: if clause of conditional statement/implication
if —p then q
Inverse: if p then q
Negation ( or —n): opposite meaning as well as opposite truth value; represented by —p
Statement: sentence that is either true or false; represented by p
RevB
Geometry A Overview
Assumptions:
We can assume the following
• Straight lines & angles are as they appear
• Collinearity of points
• Betweenneess of points
• Relative positions of points
• Adjacent r, linear pairs, supplementary Z
We cannot assume the following
• Right angles
• Congruent segments
• Congruent angles
• Relative size of segments & angles
• Perpendicular Lines
Examples: Assumptions
Use the figure below, state whether you can make the following assumptions. If not, indicate the reason.
1. ZAOC = 90°
no, cannot assume right angles
2. AOOB
ZFOB
3.
=
no, cannot assume congruency
yes, given
500
Lo2ic Table
Implication
Converse
p
q
Inverse
Contrapositive
q
p
p
q
Negation
--p (read “not p”), --p means p
Example: Converse, Inverse & Contrapositive
1.
Write the converse, inverse and contrapositive of the following true statement: “If 2 angles are right angles,
then they are congruent.” Which of your new statements are true? Which are false?
Implication: If 2 angles are right angles, then they are congruent. (True)
Converse: If 2 angles are congruent, then they are right angles. (False)
Inverse: If 2 angles are not right angles, then they are not congruent. (False)
Contrapositive: If 2 angles are not congruent, then they are not right angles. (True)
Examples: Truth Tables
Construct a truth table for each compound statement.
a. p A q
P
Q q P
T
T
F
T
F
T
A
b.
‘-g
F
T
F
TAFr=F
T ATT
FAF=F
L_LIT
FAI=
T
T
F
F
-p v—q
pv’-g_—
T F
FvF=F
F
F FT
FvTT
TvFT
T T
F
T
TvTT
F T
Properties of Equality
Reflexive Property
yçtric Property
Addition and Subtraction Properties
Multiplication and Division Properties
For every number a, a = a.
For all numbers a & b, if a = b, then b a
lfabandbcthena=c
If a = b then a + c b + c and a c b c
a
b
lfc Oanda=b,thenac=bcand—=—
—
C
L
Distributive Property
Substitution Property
C
A(b+c) = ab + ac
If a = b, then a may be replaced by b in equation
Rev B
Geometry A Overview
Properties of Inequalities:
Comparison Property
a
-
<
b, a
=
b or a> b
If a < b and b <
Transitive Property
c then a
<
c
If a> b and b> c then a> c
Addition and Subtraction Properties
Multiplication and Division Properties
If a> b then a + c> b + c and a c > b c
Ifa<bthena±c<b+canda—c<b—c
b
a
Ifc>Oanda<b,thenac<bcand—<—
—
—
C
If c> 0 and a> b, then ac> be and
C
ab
—>
C
C
—
ab
Ifc<Oanda<b,thenac>bc and—>—
C
C
ab
Ifc<Oanda>b.thenac<bcand—<—
C
--_________________________________
C
Properties
• Reflexive: AB
•
•
•
AB (any segment/angle is congruent to itself)
CD then CD AB
Symmetric: if AB
CD and CD EF then AB E EF
Transitive: if AB
if B is between A and C. then AB
Postulate:
Segment Addition
±
BC
=
AC
Examples: Algebraic Properties
State the property that justifies each statement:
Addition Property of Equality
• If x v, then x + 8 y + 8
Multiplication Property of Equality
• If x y, then 8x 8y
Transitive Property of Equality
• If a = 5 and 5 b, then a = b
Theorems and Postulates
Angles
•
•
•
•
•
•
•
•
If 2 angles are right angles, then they are congruent
Straight Angle Theorem: If 2 angles are straight angles, then they are congruent
Supplement Theorem: If 2 angles form a linear pair, then they are supplementary.
Complement Theorem: if non-common sides of 2 adjacent / s form right /, then /s are complementary
/ (or segments) then they are to each other. (Transitive Property)
to same or
If Z (or segments) are
Vertical Angle Theorem: if 2 angles are vertical angles, then they are congruent.
Same Supplements: Angles supplementary to same or congruent angles are congruent.
Same Complements: Angles complementary to same or congruent angles are congruent.
Right
Angle Theorem:
Right Angles
•
•
•
•
•
Perpendicular lines intersect to form 4 right angles
All right angles are congruent
Perpendicular lines form congruent adjacent angles
Right Angle Theorem: If 2 angles are congruent & supplementary, then each angle is a right angle
If 2 congruent angels form a linear pair, then they are right angles
4
RevB
GeQmetry A Overview
Points, Lines & Planes
• Through any 2 points, there is exactly 1 line.
• Through any 3 noncollinear points, there is exactly 1 plane.
• A line contains at least 2 points
• A plane contains at least 3 noncollinear points
• If 2 points lie in plane then entire line containing those points lies in that plane.
• If 2 lines intersect, then their intersection is exactly 1 point.
• If 2 planes intersect, then their intersection is a line
• Midpoint Theorem: If M is midpoint of AB, then AME MB
• Segment Addition Postulate: if B is between A and C, then AB + BC AC. Or if AB
is between A and C.
BC
+
=
AC, then B
General:
• Addition Property: If segments/angles are added to segments/angles, then the sums are E.
• Subtraction Property: If segment /angle is subtracted from segments/angles, the differences are
their like multiples are
• Multiplication Property: If segments/angles are
their like divisions are
• Division Property: If segments/angles are
• All radii of a circle are congruent.
,
,
Parallel, Perpendicular or Neither?
When asked to determine whether a set of lines is parallel, perpendicular or neither, you need to find the slope.
• If the slopes are the same then the lines are parallel
• If the slopes are opposite reciprocal then the lines are perpendicular
Parallel Lines & Transversal
Vocabulary:
•
•
•
•
•
•
•
Alternate Interior Angles: pair of angles in mt of figure formed by 2 lines & transversal lying on alternate
sides of transversal & having different vertices (forms a Z)
Alternate Exterior Angles: pair of angles in ext of figure formed by 2 lines & transversal lying on alternate
sides of transversal & having different vertices (forms opposite V)
Corresponding Angles: in figure formed by 2 lines & transversal, pair of angles on same side of
transversal, 1 in mt & 1 in ext having different vertices (forms F)
Exterior: outer part of figure
Interior: inner part of figure
Parallel Lines: coplanar lines that never intersect
Transversal: line that intersects 2 coplanar lines in 2 distinct points
Example:
a.
F
Which of the lines in the figure is a transversal? EF
b. Name all pairs of alternate interior angles. 3.6 and 4.5
c. Name all pairs of corresponding angles. 2,5 and 3,8 and 1,6 and 4,7
d. Name all pairs of alternate exterior angles.
B
A
c_.
.
.
3\4
C
.
6
2,7 and 1,7
F
e. Name all pairs of interior angles on same side of transversal. 3.5 and 4,6
f.
Name all pairs of exterior angles on the same side of the transversal. 2,8 and 1,7
Rev B
D
.>
______________
Geometty A Overview
Angles and Parallel Lines
Theorems:
• Alternate Interior Angles: If 2 parallel lines are cut by transversal then each pair of alternate interior
angles are congruent.
4Alt mt / )
(II
ifa
•
I
b /l/2
Alternate Exterior Angles: If 2 parallel lines are cut by transversal then each pair of alternate exterior
angles are congruent. ( Alt ext / )
a_____
I
ifaIlb/lEZ8
/
./
/8
Corresponding Angles: If 2 parallel lines are cut by transversal then each pair of corresponding angles
are congruent.
+ corresponding / )
1%
a
ifaJ!b=’/l/5
b
H/Z
(II
•
Same Side Interior Angles: If 2 parallel lines are cut by transversal then each pair of interior angles on
the same side of transversal are supplementary.
a
b
5/
•
a
if
b
=r3+z5= 180
Same Side Exterior Angles: If 2 parallel lines are cut by transversal then each pair of exterior angles on
the same side of transversal arc supplementary.
I
/
if
/
7/
•
all
all
b Zl+/7= 180
Perpendicular Transversal: In a plane, if a line is perpendicular to I of 2 parallel lines, then it is
perpendicular to the other.
if a band cIa=cIb
b
•
c
Transitive Property of Parallel Lines: If 2 lines are parallel to a
other.
a
ifa bandb c a
b
a
b
line, then they are parallel to each
llc
I
•
31
Congruent/Supplementary: If 2 parallel lines are cut by transversal then any pair of angles
x
180-x
x
180-x
180-x
x
180-x
x
Ij —>aliintZ
E,
alt ext Z
6
,
correspondingr
E,
ssi
=
180, sse
=
180
Rev B
Geometry A Overview
Examples: Angles and Parallel Lines
1. 1fcd,find Z1
c
2.IfaIb,find Li
a
100\
b4
Since c Id, alt intZ
2x + 10 3x + 5
This is a crook problem.
Li is determined by the following
1. l00+partof Li =180(SSI)
80
2. 40 part of Li (Alt Int)
40
Therefore, Li = 80 + 40 120
,
x=5
2(5)+i0=20
Since c JJd, corresponding Z
So Li = 20
,
Proving Lines are Parallel
Two lines can be proven to be parallel by:
• Alternate Interior L
• Alternate Exterior L
• Corresponding L
•
•
•
Same Side Interior (SSI) supplementary
Same Side Exterior (SSE) supplementary
Same slope
Triangle Properties
Vocabulary:
• Exterior Angle: formed by 1 side of A & extension of another side
• Interior: inside
• Remote: far away
• Remote Interior Angle: interior L of A not adjacent to given exteriorL. Interior L farthest from exterior L
Theorems:
• Angle Sum Theorem: Sum of measure of angles of a triangle is 180.
• No Choice Theorem/31d Angle Theorem: if 2 angles of triangle are to 2 angles of another triangle, then
the 3 angle of the triangles areE.
• Exterior Angle Theorem: The measure of the exterior angle of a triangle is equal to the sum of the
measures of the 2 remote interior angles.
• Exterior Angle Inequality Theorem: if L is exterior L of a triangle then its measure is greater than the
measure of its corresponding remote interiorL.
• Side Angle Theorem: longest side of a triangle is opposite the largest L in triangle.
• Angle Side Theorem: largest L of a triangle is opposite the longest side in triangle.
• Triangle Inequality Theorem: sum of lengths of any 2 sides of triangle is greater than the length of the 3”
side. (Hint: true if sum of smallest & middle> largest)
Examples: Determining if it’s a triangle
Determine whether the given measures can be the lengths of sides of a triangle.
a.2,3,4
b.6,8,l4
6+8>14
2+3>4
yes
no
7
Rev B
______________
Geometry A Overview
Examples: Determining range
d
0
r
3
f
side of triangle
Find the range for the measure of the 3 side of a triangle given the measures of 2 sides.
a. 5 and 9
To determine the lower range. subtract the 2 numbers: 9-5 4
To determine the upper range, add the 2 numbers: 9 5 = 14
Therefore, the range is 4 <x < 14
--
Examples: Side Order
Examples: Angle Order
List the sides in order from least to greatest measure.
st
1
Order angles:
ZV=28 ZU=70 ZW=82
List the angles in order from least to greatest measure.
St
1
Scm
Order sides:
p
ST=7RT=8RS= 13
\7cm
13
2 list sides associated with /
Uw, vw, uv
2 list L associated with side
ZR, ZS, rT
Classifying Triangles
Vocabulary
•
•
•
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•
•
•
Acute Triangle: all 3 angels are acute
Base: in isosceles triangle, the non congruent side
Equiangular Triangle: all 3 angles are congruent
Equilateral Triangle: all 3 sides are congruent
Hypotenuse: side across from right angle in right triangle; longest side
Isosceles Triangle: at least 2 sides are congruent
[egs: 2 sides forming right angle in right triangle; or congruent sides of isosceles tn
Obtuse Triangle: triangle with I obtuse angle & 2 acute angles
Right Triangle: triangle with 1 right angle and 2 acute angles
Scalene Triangle: all 3 sides are different lengths (no 2 sides are congruent)
Vertex Angle: angle between 2 congruent legs of isosceles triangle
Classifying Triangles by Angles
•
•
•
•
Acute: all angles are acute
Obtuse: 1 angle is obtuse
Right: 1 angle is right
Equiangular: all angles are congruent
Classifying Triangles by Sides
• Scalene: no sides are congruent
• Isosceles: at least 2 sides are congruent
• Equilateral: all 3 sides are congruent
Key Concepts:
• All equilateral triangles are isosceles but not all isosceles triangles are equilateral.
• If triangles are equilateral, then they are also equiangular and vice versa.
• If c is the length of the longest side of a triangle then
. A is obtuse
2
b
c
Aisright
+
Ifa
=
.
2
2 <c
2+b
b
c
0 If a
Aisacute
+
lfa
>
,
0
0 2
8
Rev B
Geometry A Overview
Congruent Triangles/Transformations
Vocabulary
•
•
•
•
•
Congruent Polygons: same shape/size; all pairs of corresponding parts are congruent
Congruent Triangles: all pairs of corresponding parts are congruent
Reflection: mirror image of polygon/triangle. 2 congruent triangles can be reflections of each other
Rotate: to turn polygon/triangle
Translate: push/si ide polygon/triangle
Reflection: When done on the y-axis, the x-coordinate sign changes. When done on the x-axis, the y-coordinate
sign changes.
Translation: can slide up or down, right or left or diagonally
• To right: Add number of units to x value
Add number of units to yyale
•
LLv•
To left: Subtract number of units from x value
Down: Subtract number of units from y value
Proving Triangle Congruency
Included Angle: angle between 2 sides of A
•
Included Side: sides that compose an angle
Methods for proving triangles are congruent
SSS
If there exists a correspondence
between the vertices of 2 As such
that 3 sides of I A arc to 3
sides of other A, then the 2 As
are
SAS
If there exists a correspondence
between the vertices of 2 As such
that 2 sides and the included /
of I A are to corresponding
parts of the other A, then 2 s
are.
ASA
If there exists a correspondence
between the vertices of 2 As such
that 2 /s and the included side
of I A are to the corresponding
parts of the other A, then 2 As
are.
AAS
If there exists a correspondence
between the vertices of 2 As such
that 2 Zs and the non-included
side of I A are to the
corresponding parts of the other
triangle, then 2 As are
ilL
If the hypotenuse and the leg of 1
right A are to the hypotenuse and
corresponding leg of another right A,
then As area.
LL
If legs of 1 right A are to
corresponding legs of another right
A, then As are
.
CPCTC
What is CPCTC?
• Corresponding Parts of Congruent Triangles are Congruent
When is it used?
• Only after 2 A have been proven or stated to be
Cannot be used to prove A
.
Isosceles Triangles
Angle-Side Theorems
If sides then angles: If 2 sides of a triangle areE then the angles opposite those sides are.
If angles then sides: If 2 angles of a triangle are ,then the sides opposite those angles area.
9
RevB
Geometry A Overview
Parts of Triangles
Vocabulary
• Altitude: line/segment drawn from vertex to point on opposite side making them I
• Median: line/segment drawn from I vertex of triangle to midpoint of opposite side.
Altitude of AKMO: LO
“
/
Altitude of AOLM: LN
-..—.
j
Median of ALKO: JL
a
Median of AKMO: OL
C
Equidistance Theorems
Define:
• Equidistant: distance between 2 points is equal to distance between another set of points
• Perpendicular Bisector: line that is both perpendicular to and bisects a segment. (both altitude and
median)
•
TPEEEDPB: if 2 points are each equidistant from the endpoints of a segment, then the 2 points determine
the perpendicular bisector of that segment
BD is I bis of AC
•
POPBTEE: if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints
of that segment.
If PQ is ± bisector of AB then PA PB
Proofs
Helpful hints to write a proof
• Make sure you know your previous definitions and theorems
• Use these definitions and theorems to expand the given
• Try to get to the prove statement
• Justify each step with a definition, postulate or theorem
• Mark your picture with your given and what you can observe
• Start your proof with given info. Then proceed to make conclusions from previous statements.
Indirect Proofs
Procedures for Indirect Proof
I. List the possibilities for the conclusion.
a. Your conclusion is or is not true.
2. Assume that the negation of the desired conclusion is true.
a. So the OPPOSITE of the conclusion
3. Write a “chain of reasons” until you reach an IMPOSSIBILITY or a CONTRADICTION,
a. This will be a statement that either disputes a known theorem/definitionlpostulate or your given
information.
4. State that what you assumed to start was WRONG and that the desired conclusion then must be true.
10
RevB
Geometry A Overview
Quadrilaterals
Parallelogram
•
•
•
•
•
Opposite sides are parallel
Opposite sides are
Opposite angles are
Consecutive angles are
supplementary
Diagonal bisect each other
Rectangle
•
•
•
•
All properties of parallelogram
All angles are right angles
Diagonals are
Diagonals divide rectangle into
isosceles triangles.
Trapezoid
Kite
•
•
•
•
2 disjoint pairs of consecutive
sides are
Diagonals are perpendicular
1 diagonal is perpendicular
bisector of the other
1 of diagonals bisects a pair of
opposite angles
1 pair of opposite angles is
4.
•
1 pair of parallel lines
I
Isosceles Trapezoid
Rhombus
•
•
•
•
•
•
all properties of parallelogram
all properties of kite
all sides are
diagonals bisect angles
diagonals are perpendicular
bisectors of each other
Diagonals divide rhombus into 4
right triangles.
•
•
•
•
•
•
legs are congruent
bases are parallel
lower base angles are
upper base angles are
diagonals are congruent
any lower base angle is
supplementary to any upper
base angle
Square
•
•
•
all propees of rectangle
all properties of rhombus
diagonals form 4 isosceles right
triangles (45-45-90 triangles)
11
RevB