Cogruence

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Cogruence

  1. 1. Math presentation About “ CONGRUENCE ” By : Rizaldi Pahlevi and Abel Agusta Banuboro
  2. 2. 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 <ul><li>The conditions for the congruence of two figures are: </li></ul><ul><li>All the corresponding sides are equal in length, and </li></ul><ul><li>All the corresponding angles are equal in measure </li></ul>
  3. 3. 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 <ul><li>Thus, the rectangles ABCD and EFGH are similar and have </li></ul><ul><li>the following properties : </li></ul><ul><li>All the corresponding sides are proportional. </li></ul><ul><li>All the corresponding angles are equal in measure. </li></ul><ul><li>Because ABCD and EFGH are similar, we can conclude if </li></ul><ul><li>point 1 and 2 is “The conditions for similarity of two figures” </li></ul>
  4. 4. Congruent Triangles
  5. 5. Corresponding parts of congruent triangles <ul><li>Triangles that are the same size and shape are congruent triangles . </li></ul><ul><li>Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent. </li></ul>
  6. 6. 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 ~
  7. 7. 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 ~
  8. 8. Properties of Triangle Congruence <ul><li>Congruence of triangles is reflexive , symmetric, and transitive. </li></ul><ul><li>REFLEXIVE </li></ul>K J L K J L Δ JKL = Δ JKL ~
  9. 9. Properties of Triangle Congruence <ul><li>Congruence of triangles is reflexive, symmetric , and transitive. </li></ul><ul><li>SYMMETRIC </li></ul>K J L Q P R If Δ JKL = Δ PQR, then Δ PQR = Δ JKL. ~ ~
  10. 10. Properties of Triangle Congruence <ul><li>Congruence of triangles is reflexive, symmetric, and transitive . </li></ul><ul><li>TRANSITIVE </li></ul>K J L Q P R If Δ JKL = Δ PQR, and Δ PQR = Δ XYZ, then Δ JKL = Δ XYZ. ~ ~ ~ Y X Z
  11. 11. CONDITIONS for CONGRUENCE of TWO TRIANGLE
  12. 12. Side-Side-Side (SSS) <ul><li>AB  DE </li></ul><ul><li>BC  EF </li></ul><ul><li>AC  DF </li></ul> ABC   DEF E D F B A C
  13. 13. Side-Angle-Side (SAS) <ul><li>AB  DE </li></ul><ul><li> A   D </li></ul><ul><li>AC  DF </li></ul> ABC   DEF B A C E D F included angle
  14. 14. The angle between two sides Included Angle  G  I  H
  15. 15. 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
  16. 16. Angle-Side-Angle (ASA) <ul><li> A   D </li></ul><ul><li>AB  DE </li></ul><ul><li> B   E </li></ul> ABC   DEF B A C E D F included side
  17. 17. The side between two angles Included Side GI HI GH
  18. 18. Name the included angle :  Y and  E  E and  S  S and  Y Included Side YE ES SY S Y E
  19. 19. Angle-Angle-Side (AAS) <ul><li> A   D </li></ul><ul><li> B   E </li></ul><ul><li>BC  EF </li></ul> ABC   DEF B A C E D F Non-included side
  20. 20. Warning: No SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!
  21. 21. Warning: No AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT
  22. 22. CONDITIONS for SIMILARITY of TWO TRIANGLE
  23. 23. All the corresponding sides of the two triangles are PROPORTIONAL 1. A C B P R Q AB PQ BC QR AC PR = =
  24. 24. 2. Two angles of one triangle are equal in measure to two corresponding angles of the other triangle. A b c g h i
  25. 25. 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
  26. 26. 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
  27. 27. 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

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