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picture1_Geometry Pdf 167653 | 042115 Geometry Regents Summary


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File: Geometry Pdf 167653 | 042115 Geometry Regents Summary
things to know for the geometry regents exam o angles inside triangles add to 180 o angles inside quadrilaterals add to 360 angles inside any polygon with n sides add ...

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                                               Things to Know for the 
                                           Geometry Regents Exam 
                                                                                                                            o  
                                                                  Angles INSIDE Triangles: add to 180  
                                                                                                                                      o
                                                                  Angles INSIDE Quadrilaterals:  add to 360    
                                                                   Angles INSIDE any polygon with “n” sides:  add to 180o(n – 2) 
                                                                            If polygon is regular (all sides and all angles are congruent) then 
                                                                                                                                        ೚
                                                                                                                                   ଵ଼଴ ሺ௡ିଶሻ
                                                                                      EACH angle inside the polygon measures:             ௡        
                                                                   Angles OUTSIDE any polygon:  add to 3600  where each exterior angle in 
                                                                                                           ଷ଺଴
                                                                        a regular polygon measures   ௡  
                             ANGLES          Polygons to Know:  
                                                                        Name          Triangle  Quadrilateral  Pentagon Hexagon Octagon  Decagon 
                                                                     # of sides            3 4 5 6 8 10 
                                                                                                                                             o 
                                                                   Complementary Angles: two angles that add to 90
                                                                                                                                              o
                                                                        Supplementary Angles: two angles that add to 180  
                                                                                                                             o
                                                                        Linear Pair:  two angles that add to 180  and are adjacent (form a line) 
                                                                                            Ex)                110       70
                         
                                                                   Vertical Angles:  (the angles opposite one another that are formed when  
                                                                        two lines intersect) VERTICAL ANGLES ARE CONGRUENT. 
                                                                                            Ex) 
                                                                                                                 110     70
                                                                                                            70
                                                                                                                 110
                          
                                                                   Note:  When finding the Volume of a solid, “B” stands for the Area of Base. 
                                                                                             2                                                                   ଵ
                                                                      Square:  A = s      Rectangle:  A = LW        Triangle:  A =  ܾ݄ 
                                  AREA                        Circle:     A = ߨݎଶ        Trapezoid:  A = ଵ݄ሺܾ ൅ܾሻ                                                ଶ
                                                                                                                        ଶ              ଶଶ     ଵ      ଶ
                                                                   Distance Formula:  ݀ൌ	√ሺݔ െݔሻ  +ሺݕ െݕሻ   
                                                                                                             ଶ       ଵ         ଶ      ଵ             Distance:  used to prove 
                                                                                                          ௫ 	ା	௫     ௬ 	ା	௬
                                                Midpoint Formula: ܯൌሺ	భଶ మ	, భଶ మ	ሻ                                                                       CONGRUENT 
                                                                                                    ௬ 	ି		௬                                        Midpoint:  used to prove 
                         COORDINATE      Slope Formula:  ݉ൌ	మ                                               భ                                              BISECTING 
                                                                                                    ௫మ	ି	௫భ                                           Slope:  used to prove 
                            GEOMETRY     Equation of a Circle:  where r = radius                                                                           PARALLEL  
                                                                                                   ଶ      ଶ     ଶ                                        (equal slopes) 
                              including                    centered at origin:  ݔ ൅ݕ ൌݎ                                                                        or 
                                                                                                         2            2     2                           PERPENDICULAR  
                                  CIRCLES                    centered at (h, k):  (x – h)  + (y – k)  = r                                            (Neg. Reciprocal Slopes) 
                                                     Ex)  What is the center and radius of this circle: 
                                                                                         2            2
                                                                                      (x – 3)  + (y + 5)  = 16  ? 
                                                                                       
                         
                                                                                         Center:  (+3, -5) 
                                                                                                                  
                                                                                                          Radius:   16 = 4                Notice:  Change the signs of x and y to find center 
                                                                                             √                                                                   2   
                                                                                                                       If no number is written (as in x ), then use 
                                                                                                                  
                                                                                                                       zero. Also, notice that the number after  
                                                                                                                  
                                                                                                                       the equal sign is the radius after being  
                                                                                                                       squared. 
                                                                    
                         
                                                                                                                                                                                 1 
                                                       
                                                                                               Central Angle:        Inscribed Angle:               Vertical Angles:      
                                                                                                             EQUAL to the arc             HALF the arc                      ADD the arcs then divide by 2 
                                                                                                                                                                               
                                                                                                                                                                                                                                                                                                                                                                                                 ଵ଻଴ା଻଴
                                                                                                                                                                                                                                                                                                                                                                                        x =             ଶ         
                                                                                                                                                                                                                                                                                                                                                                                                             0 
                                                                                                                                                                                                                                                                                                                                                                                          x = 120
                                                             ANGLES                                                                                                                                                                                                            o
                                                                                                                                                                                                                                                                       ½ x 
                                                                                           in                                                                                                            o                                                             80
                                                                                                                                                                                                 80
                                                           CIRCLES                                                                                                                                x                                                                           x = ½(80) 
                                                                                                                                                                                               o                                                                                                o 
                                                                                                                                                                               x = 80                                                                                           x = 40
                                                                                                     Angle OUTSIDE Circle:                     Tangent/Chord Angle: 
                                                                                                          SUBTRACT the arcs then divide by 2                        HALF the arc                         
                                                                                                                                                     
                                                       
                                                       
                                                         x = ½(120) 
                                                                                                                                                                                                                                                 ଼଴ିଶ଴                                                                                                                                               o 
                                                                                                                                                                                                                                        x =           ଶ                                                                                                                              x = 60
                                                                                                                                                                                                                                                          o 
                                                                                                                                                                                                                                           x = 30
                                                                                                      Intersecting Chords:                         Two Secants:  
                                                                                                    (LEFT)(RIGHT) = (LEFT)(RIGHT)       (WHOLE) (OUTER) = (WHOLE) (OUTER) 
                                                                                                                                                                                               x
                                                                                                                                                                                                                                                                                                                                                       (x + 5)(5) = (10)(6) 
                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                                                                                                                                                                                                                                                                          5x + 25 =  60 
                                                         SEGMENTS                                                                                                                                                                      x ∙ 2 = 3 ∙ 4                                                                                                               5x = 35 
                                                                                                                                                                                                                                          x = 6                                                                                                                      x = 7 
                                                                          in 
                                                         CIRCLES            Secant/Tangent:                                Two Tangents: 
                                                                                                                                                                                                                                                               2
                                                                                                   (WHOLE)(OUTER) = (TANGENT)              Are CONGRUENT to one another 
                                                                                                                                                                                                                                                                                                                                              17 
                                                                                                                                                                                                                                                                                                                                            x                              x = 17
                                                                                                                                                                                                                                                                                                                                                                       
                                                                                                                                                   Tangent/Diameter:     Chord ٣ Diameter: 
                                                                                                                                                     are Perpendicular       will BISECT the chord   
                                                                                                                                                                                                                  2
                                                                                                                                                                                                               
                                                                                                                                                                                     (12)(5) = x
                                                                                                                                                                                                              2
                                                                                                                                                                                          60 = x  
                                                                                                                                                                                              60 = x 
                                                                                                                                                                                          √
                                                                                                                                                                                         2√15= x                                                                              Congruent Segments:  If segments are ≅,  
                                                                                                  Parallel Segments: If 2 segments                                                                                                                                            the arcs they intercept are also  ≅. 
                                                                                                           are parallel, then ARCS BETWEEN 
                                                                                                           are congruent.                                                                                                     If AB∥CD, 
                                                                                                                                                                                                                                            ෢ ෢
                                                                                                                                                                                                                              then ܤܥ = ܣܦ 
                                                                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                                                                                                                                           
                                                                                                                                                                                                                                                                                                                                                                                                                      2 
                                                                       Parallelogram:  opposite sides congruent and parallel              
                                                                                                                          opposite angles congruent            
                                                                                               consecutive ∢ݏ	supplementary        
                                                                                                     diagonals BISECT each other                          
                                                                                               
                                                                                                                                                                                          
                                                                                                                                                                             a + b =  180
                           
                                                               Rectangle:                                                 Rhombus 
                                                                            all 90o ∢ݏ                                                    all sides ≅ 
                                                                            diagonals ≅                           diagonals ٣     
                                                                                                                                                                   diagonals BISECT ∢ݏ  
                                                                                                                 
                           
                           
                                                     
                          QUADRILATERALS                            Square: 
                                       including                                                                                                             
                                PARALLELOGRAM                                                                                   ALL Properties ABOVE 
                                      FAMILY 
                                            & 
                           TRAPEZOID FAMILY 
                           
                                                         Trapezoid: only ONE pair of opposite sides are PARALLEL                                                           a     >> d
                                                                       Angles:   a + b = 180o, c + d = 180o 
                                                                                                                                                                     b        >>            c
                                                                                                                          if non-parallel sides 
                                                                                                                            are CONGRUENT 
                           
                                                                               Isosceles Trapezoid: 
                                                                                                          
                                                                                                        Upper Base Angles ≅    
                                                                                                                                                              Lower Base Angles ≅    o 
                                                                                                                                                                    1 Upper + 1 Lower = 180
                                                                                                        Diagonals ≅ 
                           
                           
                                                         Proving a Parallelogram:  find  DISTANCE of all 4 sides and show 
                                                                             opposite sides are CONGRUENT (because they have the same distance). 
                                                                            Proving a Rectangle:  find DISTANCE of all 4 sides AND the 2 diagonals 
                                                                            and show that opposite sides are CONGRUENT and the diagonals are also. 
                           COORDINATE         Proving a Rhombus:  find DISTANCE of all 4 sides and show that ALL  
                               GEOMETRY               sides are CONGRUENT (because they have the same distance). 
                                  PROOFS                     Proving a Square: find DISTANCE of all 4 sides AND the 2 diagonals 
                                                                            and show that ALL sides are CONGRUENT and the diagonals are also. 
                                                                            Proving a Trapezoid:  find SLOPE of all 4 sides and show that one pair of  
                                                                            opposite sides is PARALLEL (b/c they have the same slope) and the other  
                                                                            pair is NOT PARALLEL (b/c they have different slopes). 
                                                                            Proving an Isosceles Trapezoid: First, prove it’s a trapezoid (see above) 
                                                                            then find DISTANCE of the NON-PARALLEL sides and show they are ≅. 
                                                                                  So, when do we use the Midpoint Formula in Proofs?  Only if we’re  
                                                                            asked to prove that segments BISECT each other (same midpoint → bisect). 
                                                                                                                                                                                                 3 
                                                                    Types of Triangles:  
                                                                      By SIDES → Scalene:  no ≅ sides      By ANGLES→ Acute:  all 3 acute ∢ݏ 
                         TRIANGLE                           Isosceles: 2 ≅ sides                             Right: 1 right ∢ (2 acute)    
                          TYPES                        Equilateral: 3 ≅ sides                         Obtuse: 1 obtuse (2 acute)    
                                                   Isosceles Triangle: 2 ≅ sides called LEGS; other side is BASE.  Angles  
                                                                      opposite legs are ≅ (BASE ANGLES); other angle is VERTEX.               o
                                                          Equilateral Triangle:  all sides ≅, all angles ≅ (each angle measures 60 ) 
                                                                      
                                                                Median:  BISECTS the opposite SIDE (intersects at midpoint of opp. side)   
                                                                                                    MEDIAN
                     
                                                   Altitude:  meets the opposite side and forms a right angle (٣) 
                                                                                                  ALTITUDE
                     
                                                   Angle Bisector:  BISECTS the ANGLE from where it was drawn  
                                                                                1  2           ANGLE BISECTOR    ܪ݁ݎ݁:		∡1	 ≅ 	∡2
                                                    
                                                   Perpendicular Bisector:  (1) BISECTS the opposite SIDE and (2) forms 
                                                                     a right angle with opposite side (Notice:  It does NOT have to come from opposite ∡ሻ 
                                                              
                     
                        SEGMENTS          Points of CONCURRENCE:  since each triangle has 3 of each of the above 
                                      IN                          line segments, the point where these lines intersect is called… 
                       TRIANGLES                                              Name of Point           Intersection of 
                                                                                                        the three…              To remember how 
                                                                                                                                 these “pair off”: 
                                                                                 CENTROID Medians  Alphabetize the 
                                                                                                                                names of 3 points , 
                                                                              CIRCUMCENTER  Perp. Bisectors                     then line them up 
                                                                                 INCENTER Angle Bisectors  by remembering 
                                                                                                                                 “My Parents Are 
                                                                              ORTHOCENTER Altitudes                                 ALiens.” 
                                                                Centroid: 
                                                                Will always be located inside the triangle. 
                                                                     Divides into 2:1 ratio (section near vertex 
                                                                        is twice as long as section near midpt). 
                                                                  
                     
                     
                     
                                                                    Circumcenter: 
                                                                    Will be inside if triangle is ACUTE.  
                                                                    Will be outside if triangle if OBTUSE. 
                                                                    Will be on triangle if triangle is RIGHT.  
                     
                     
                                                                                                                                                    4 
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...Things to know for the geometry regents exam o angles inside triangles add quadrilaterals any polygon with n sides if is regular all and are congruent then each angle measures outside where exterior in a polygons name triangle quadrilateral pentagon hexagon octagon decagon of complementary two that supplementary linear pair adjacent form line ex vertical opposite one another formed when lines intersect note finding volume solid b stands area base square s rectangle lw circle trapezoid distance formula used prove midpoint coordinate slope bisecting equation r radius parallel equal slopes including centered at origin or perpendicular circles h k x y neg reciprocal what center this notice change signs find no number written as use zero also after sign being squared central inscribed arc half arcs divide by...

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