Download Introduction to Geometry

Geometry A Overview
General Formulas
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Distance Formula: Given P=(x1,y1) and Q = (x2,y2), distance is d =
( x1  x2 ) 2 ( y1  y2 ) 2
 x  x2 y1  y 2 
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Midpoint Formula: Given P=(x1,y1) and Q = (x2,y2), midpoint is  1
,
2 
 2
y  y2
Slope Formula: Given P=(x1,y1) and Q = (x2,y2), slope is m = 1
x1  x2
Symbols
Symbol
≠

≥
≤
¬, ~


Meaning
Not equal
Congruent
Greater than or equal to
Less than or equal to
Not
And
Or
Symbol
∆



∩


Meaning
Triangle
Angle
Perpendicular
Union
Intersection
Implies
If and only if
Symbol





ε
ε
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
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Angle (  ): formed 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
Ray( 
 ): begins at an endpoint & then extends infinitely far in only 1 direction.
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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
English Word Algebraic Translation
Complement
90 –x
Difference
Equal
=
Greater than
+, >
Increased by
+
Less than
-, <
English Word
Number
Opposite of a number
Product
Sum
Supplement
Algebraic Translation
N
-N
*
+
180 – x
Example: Angle Measures
1. Find the measures of 2 supplementary angels if the measure of 1 angle is 6 less than 5 times the measure of
the other angle.
180 –x = 5x - 6; angles149, 31
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Rev B
Geometry A Overview
Angle Relationships
Vocabulary
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Acute Angle: between 0º and 90º
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 90º
Linear Pair: pair of adjacent angels whose non-common sides are opposite rays.
Obtuse Angle: between 90º and 180º
Perpendicular (  ): lines that form 90º angles. Intersect to form congruent adjacent angles; rt angle symbol
indicates lines are perpendicular
Right Angle: 90º
Straight Angle: 180º
Supplementary Angles: 2 angles whose measures have sum of 180º
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 &  CEB,  AEB &  CED
3. Linear Pair:  DEA &  BEA,  DEC &  BEC
4. Given the diagram to the right, identify all angle relationships.
  1 and  5 are exterior angles
  1 sup  2 ,  1 +  2 =180,  1 &  2 are linear pair
  4 sup  5 ,  4 +  5 =180,  4 &  5 are linear pair
  2 +  3 +  4 = 180
  1 =  3 +  4,  1 >  3,  1 >  4
  5 =  2 +  3 ,  5 >  2,  5 > 3
Reasoning & Proofs
Vocabulary
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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 (  ): compound statement formed by joining 2 or more statements with word and; true when
both are true
Contrapositive: if p then q  if ~q then ~p
Converse: statement associated with if p then q having form if q then p
Disjunction (  ): 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
Inverse: if p then q  if ~p then ~q
Negation (~ or  ): opposite meaning as well as opposite truth value; represented by ~p
Statement: sentence that is either true or false; represented by p
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Rev B
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  , linear pairs, supplementary 
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.  AOC = 90º
no, cannot assume right angles
2. AO  OB
3.
 FOB = 50º
no, cannot assume congruency
yes, given
Logic Table
Implication p  q
Converse
qp
Inverse
Contrapositive
~p  ~q
~q  ~p
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  ~q
P Q ~q P  ~q
T T F
T F=F
T F T
T T=T
F T F
FF = F
F F T
F T=F
b. ~p  ~q
p q ~p
T T F
T F F
F T T
F F T
~q
F
T
F
T
~p  ~q
F F=F
F T=T
T F = T
T T=T
Properties of Equality
Reflexive Property
Symmetric Property
Transitive Property
Addition and Subtraction Properties
Multiplication and Division Properties
Distributive Property
Substitution Property
For every number a, a = a.
For all numbers a & b, if a = b, then b = a
If a = b and b = c then a = c
If a = b then a + c = b + c and a – c = b – c
a b
If c  0 and a = b, then ac = bc and =
c c
A(b+c) = ab + ac
If a = b, then a may be replaced by b in equation
Properties of Inequalities:
3
Rev B
Geometry A Overview
a < b, a = b or a > b
If a < b and b < c then a < c
If a > b and b > c then a > c
If a > b then a + c > b + c and a – c > b – c
Addition and Subtraction Properties
If a < b then a + c < b + c and a –c < b – c
Multiplication and Division Properties
a b
If c > 0 and a < b, then ac < bc and <
c c
a b
If c > 0 and a > b, then ac > bc and >
c c
a b
If c < 0 and a < b, then ac > bc and >
c c
a b
If c < 0 and a > b, then ac < bc and <
c c
Comparison Property
Transitive Property
Properties
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Reflexive: AB  AB (any segment/angle is congruent to itself)
Symmetric: if AB  CD then CD  AB
Transitive: if AB  CD and CD  EF then AB  EF
Segment Addition Postulate: if B is between A and C, then AB + BC = AC
Examples: Algebraic Properties
State the property that justifies each statement:
 If x = y, then x + 8 = y + 8
Addition Property of Equality
 If x = y, then 8x = 8y
Multiplication Property of Equality
 If a = 5 and 5 = b, then a = b
Transitive Property of Equality
Theorems and Postulates
Angles
 Right Angle Theorem: 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
 If  (or segments) are  to same or   (or segments) then they are  to each other. (Transitive Property)
 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 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
Points, Lines & Planes
 Through any 2 points, there is exactly 1 line.
4
Rev B
Geometry A Overview
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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 AM  MB
Segment Addition Postulate: if B is between A and C, then AB + BC = AC. Or if AB + BC = AC, then B
is between A and C.
General:
 Addition Property: If  segments/angles are added to  segments/angles, then the sums are  .
 Subtraction Property: If  segment /angle is subtracted from  segments/angles, the differences are  .
 Multiplication Property: If segments/angles are  , their like multiples are  .
 Division Property: If segments/angles are  , their like divisions 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:
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Alternate Interior Angles: pair of angles in int 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 int & 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. Which of the lines in the figure is a transversal? EF
E
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.
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
Angles and Parallel Lines
Theorems:
5
Rev B
Geometry A Overview
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Alternate Interior Angles: If 2 parallel lines are cut by transversal then each pair of alternate interior
angles are congruent. (║Alt int   )
a
if a ║ b  1  2
1
b
2
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Alternate Exterior Angles: If 2 parallel lines are cut by transversal then each pair of alternate exterior
angles are congruent. (║Alt ext   )
a
1
if a ║ b  1  8
b
8
 Corresponding Angles: If 2 parallel lines are cut by transversal then each pair of corresponding angles
are congruent. (║corresponding   )
1
a
if a ║ b  1  5
5
b
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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.
3
5
a
b
if a ║ b  3 5  180
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Same Side Exterior Angles: If 2 parallel lines are cut by transversal then each pair of exterior angles on
the same side of transversal are supplementary.
a
1
if a ║ b  1 7  180
b
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 Perpendicular Transversal: In a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is
perpendicular to the other.
a
if a ║ b and ca  cb
b
c
Transitive Property of Parallel Lines: If 2 lines are parallel to a 3rd line, then they are parallel to each
other.
a
if a ║ b and b ║ c  a ║c
b
c
 Congruent/Supplementary: If 2 parallel lines are cut by transversal then any pair of angles
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a
b
x 180-x
180-x x
x 180-x
180-x x
||  alt int   , alt ext   , corresponding   , ssi = 180, sse = 180
Examples: Angles and Parallel Lines
1. If c || d, find 1
2. If a || b, find 1
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Rev B
Geometry A Overview
c
1
d
2x + 10
a
3x + 5
100
b
Since c ||d, alt int   ,
2x + 10 = 3x + 5
x=5
2 (5) + 10 = 20
1 c
40
This is a crook problem.
1 is determined by the following
1. 100 + part of 1 = 180 (SSI)
= 80
2. 40  part of 1 (Alt Int)
= 40
Therefore, 1 = 80 + 40 = 120
Since c ||d, corresponding   ,
So 1 = 20
Proving Lines are Parallel
Two lines can be proven to be parallel by:
 Alternate Interior  
 Alternate Exterior  
 Corresponding  
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Same Side Interior (SSI) supplementary
Same Side Exterior (SSE) supplementary
Same slope
Triangle Properties
Vocabulary:
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Exterior Angle: formed by 1 side of ∆ & extension of another side
Interior: inside
Remote: far away
Remote Interior Angle: interior  of ∆ not adjacent to given exterior. Interior  farthest from exterior 
Theorems:
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Angle Sum Theorem: Sum of measure of angles of a triangle is 180.
No Choice Theorem/3rd Angle Theorem: if 2 angles of triangle are  to 2 angles of another triangle, then
the 3rd angle of the triangles are  .
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  is exterior  of a triangle then its measure is greater than the
measure of its corresponding remote interior.
Side Angle Theorem: longest side of a triangle is opposite the largest  in triangle.
Angle Side Theorem: largest  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 3rd
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, 14
2+3>4
6 + 8 > 14
yes
no
Examples: Determining range of 3rd side of triangle
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Rev B
Geometry A Overview
Find the range for the measure of the 3rd 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.
1st Order angles:
V=28 U=70 W=82
List the angles in order from least to greatest measure.
1st Order sides:
ST =7 RT = 8 RS = 13
2nd list  associated with side
R, S, T
2nd list sides associated with 
UW, VW, UV
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
Legs: 2 sides forming right angle in right triangle; or congruent sides of isosceles tri
Obtuse Triangle: triangle with 1 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
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Acute: all angles are acute
 Obtuse: 1 angle is obtuse
 Right: 1 angle is right
 Equiangular: all angles are congruent
Classifying Triangles by Sides
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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
o If a2 + b2 > c2, ∆ is acute
o If a2 + b2 = c2, ∆ is right
o If a2 + b2 < c2, ∆ is obtuse
Congruent Triangles/Transformations
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Rev B
Geometry A Overview
Vocabulary
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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/slide 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
 Up: Add number of units to the y value
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 To left: Subtract number of units from x value
 Down: Subtract number of units from y value
Proving Triangle Congruency
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Included Angle: angle between 2 sides of ∆
Included Side: sides that compose an angle
Methods for proving triangles are congruent
SSS
SAS
ASA
If there exists a correspondence
If there exists a correspondence
If there exists a correspondence
between the vertices of 2 ∆s such
between the vertices of 2 ∆s such
between the vertices of 2 ∆s such
that 3 sides of 1 ∆ are  to 3
that 2 sides and the included 
that 2 s and the included side
sides of other ∆, then the 2 ∆s
of 1 ∆ are  to corresponding
of 1 ∆ are  to the corresponding
are  .
parts of the other ∆, then 2 ∆s
parts of the other ∆, then 2 ∆s
are  .
are  .
AAS
If there exists a correspondence
between the vertices of 2 ∆s such
that 2 s and the non-included
side of 1 ∆ are  to the
corresponding parts of the other
triangle, then 2 ∆s are  .
HL
If the hypotenuse and the leg of 1
right ∆ are  to the hypotenuse and
corresponding leg of another right ∆,
then ∆s are  .
LL
If legs of 1 right ∆ are  to
corresponding legs of another right
∆, then ∆s are  .
CPCTC
What is CPCTC?
 Corresponding Parts of Congruent Triangles are Congruent
When is it used?
 Only after 2 ∆ have been proven or stated to be  . Cannot be used to prove ∆ 
Isosceles Triangles
Angle-Side Theorems
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If sides then angles: If 2 sides of a triangle are  , then the angles opposite those sides are  .
If angles then sides: If 2 angles of a triangle are  , then the sides opposite those angles are  .
Parts of Triangles
Vocabulary
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Rev B
Geometry A Overview
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Altitude: line/segment drawn from vertex to point on opposite side making them 
 Median: line/segment drawn from 1 vertex of triangle to midpoint of opposite side.
Altitude of ∆KMO: LO
Altitude of ∆OLM: LN
Median of ∆LKO: JL
Median of ∆KMO: OL
Equidistance Theorems
Define:
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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)
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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  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
1. 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/definition/postulate or your given
information.
4. State that what you assumed to start was WRONG and that the desired conclusion then must be true.
Quadrilaterals
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Rev B
Geometry A Overview
Parallelogram
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Opposite sides are parallel
Opposite sides are 
Opposite angles are 
Consecutive angles are
supplementary
Diagonal bisect each other
Kite
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Rectangle
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All properties of parallelogram
All angles are right angles
Diagonals are 
Diagonals divide rectangle into
isosceles triangles.
Trapezoid
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 
Rhombus
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1 pair of parallel lines
Isosceles Trapezoid
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.
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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
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all properties of rectangle
all properties of rhombus
diagonals form 4 isosceles right
triangles (45-45-90 triangles)
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Rev B