Cogruence
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Cogruence

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Cogruence Cogruence Presentation Transcript

  • Math presentation About “ CONGRUENCE ” By : Rizaldi Pahlevi and Abel Agusta Banuboro
  • A B C D K L M N Look at the figure ! From the figure we know if : AB = KL  A =  K BC = LM and  B =  L CD = MN  C =  M DA = NK  D =  N } congruent The conclusion is “If there two plane which are perfectly coincident are called two congruent figures” Congruence of two figures
    • The conditions for the congruence of two figures are:
    • All the corresponding sides are equal in length, and
    • All the corresponding angles are equal in measure
  • Similarity of two figures A B C D e f g h Look at figure below,! x y 2x 2y Thus, the ratios of the corresponding sides are equal : EF : AB = GH : CD = EH : AD = FG : CB = 2 : 1 From the figure we know if :  A =  E = 90 0  B =  F = 90 0  C =  G = 90 0  D =  H = 90 0
    • Thus, the rectangles ABCD and EFGH are similar and have
    • the following properties :
    • All the corresponding sides are proportional.
    • All the corresponding angles are equal in measure.
    • Because ABCD and EFGH are similar, we can conclude if
    • point 1 and 2 is “The conditions for similarity of two figures”
  • Congruent Triangles
  • Corresponding parts of congruent triangles
    • Triangles that are the same size and shape are congruent triangles .
    • Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent.
  • Corresponding parts of congruent triangles If Δ ABC is congruent to Δ XYZ , then vertices of the two triangles correspond in the same order as the letter naming the triangles. A C B X Z Y Δ ABC = Δ XYZ ~
  • Corresponding parts of congruent triangles This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles. A C B X Z Y Δ ABC = Δ XYZ ~
  • Properties of Triangle Congruence
    • Congruence of triangles is reflexive , symmetric, and transitive.
    • REFLEXIVE
    K J L K J L Δ JKL = Δ JKL ~
  • Properties of Triangle Congruence
    • Congruence of triangles is reflexive, symmetric , and transitive.
    • SYMMETRIC
    K J L Q P R If Δ JKL = Δ PQR, then Δ PQR = Δ JKL. ~ ~
  • Properties of Triangle Congruence
    • Congruence of triangles is reflexive, symmetric, and transitive .
    • TRANSITIVE
    K J L Q P R If Δ JKL = Δ PQR, and Δ PQR = Δ XYZ, then Δ JKL = Δ XYZ. ~ ~ ~ Y X Z
  • CONDITIONS for CONGRUENCE of TWO TRIANGLE
  • Side-Side-Side (SSS)
    • AB  DE
    • BC  EF
    • AC  DF
     ABC   DEF E D F B A C
  • Side-Angle-Side (SAS)
    • AB  DE
    •  A   D
    • AC  DF
     ABC   DEF B A C E D F included angle
  • The angle between two sides Included Angle  G  I  H
  • Name the included angle: Y E and E S E S and Y S Y S and Y E Included Angle S Y E  E  S  Y
  • Angle-Side-Angle (ASA)
    •  A   D
    • AB  DE
    •  B   E
     ABC   DEF B A C E D F included side
  • The side between two angles Included Side GI HI GH
  • Name the included angle :  Y and  E  E and  S  S and  Y Included Side YE ES SY S Y E
  • Angle-Angle-Side (AAS)
    •  A   D
    •  B   E
    • BC  EF
     ABC   DEF B A C E D F Non-included side
  • Warning: No SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!
  • Warning: No AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT
  • CONDITIONS for SIMILARITY of TWO TRIANGLE
  • All the corresponding sides of the two triangles are PROPORTIONAL 1. A C B P R Q AB PQ BC QR AC PR = =
  • 2. Two angles of one triangle are equal in measure to two corresponding angles of the other triangle. A b c g h i
  • 3. A b c D E F An angle of one triangle is equal in measure to an angle of the other triangle, and the sides which include the equal angle of both triangles are proportional
  • The formulas for a right triangle with altitude on the hypotenuse A B C AD 2 = BD X CD AB 2 = BD X BC D AC 2 = CD X CB
  • The formulas for a triangle containing a line parallel to one of its sides A B C D E a b c d e f = = = = = = > > Cd ca Ce cb De ab a a+b C c+d e f a b c d a c b d