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G.6

Proving Triangles Congruent. G.6. Visit www.worldofteaching.com For 100’s of free powerpoints. F. B. A. C. E. D. The Idea of Congruence. Two geometric figures with exactly the same size and shape. How much do you need to know. . . . . . about two triangles

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G.6

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  1. Proving Triangles Congruent G.6 Visit www.worldofteaching.com For 100’s of free powerpoints.

  2. F B A C E D The Idea of Congruence Two geometric figures with exactly the same size and shape.

  3. How much do you need to know. . . . . . about two triangles to prove that they are congruent?

  4. B A C E F D Corresponding Parts Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. • AB DE • BC EF • AC DF •  A  D •  B  E •  C  F ABC DEF

  5. SSS SAS ASA AAS HL Do you need all six ? NO !

  6. B A C Side-Side-Side (SSS) If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Side E Side F D Side • AB DE • BC EF • AC DF ABC DEF The triangles are congruent by SSS.

  7. Included Angle The angle between two sides • GHI • H • GIH • I • HGI • G This combo is called side-angle-side, or just SAS.

  8. E Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE YES or E YSE or S EYS or Y The other two angles are the NON-INCLUDED angles.

  9. Side-Angle-Side (SAS) If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. included angle B E Side F A Side C D • AB DE • A D • AC DF Angle ABC DEF The triangles are congruent by SAS.

  10. Included Side The side between two angles GI GH HI This combo is called angle-side-angle, or just ASA.

  11. E Y S Included Side Name the included side: Y and E E and S S and Y YE ES SY The other two sides are the NON-INCLUDED sides.

  12. Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. included side B E Angle Side F A C D Angle • A D • AB  DE • B E ABC DEF The triangles are congruent by ASA.

  13. Angle-Angle-Side (AAS) If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. B Non-included side E Angle F Side A C D Angle • A D • B E • BC  EF ABC DEF The triangles are congruent by AAS.

  14. Warning: No SSA Postulate There is no such thing as an SSA postulate! Side Angle Side The triangles are NOTcongruent!

  15. Warning: No SSA Postulate There is no such thing as an SSA postulate! NOT CONGRUENT!

  16. BUT: SSA DOES work in one situation! If we know that the two triangles are right triangles! Side Side Side Angle

  17. We call this HL, for “Hypotenuse – Leg” Remember! The triangles must be RIGHT! Hypotenuse Hypotenuse Leg RIGHT Triangles! These triangles ARE CONGRUENT by HL!

  18. Hypotenuse-Leg (HL) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Right Triangle Leg Hypotenuse • ABHL • CB  GL • C andG are rt.  ‘s ABC DEF The triangles are congruent by HL.

  19. Warning: No AAA Postulate There is no such thing as an AAA postulate! Different Sizes! E Same Shapes! B A C F D NOT CONGRUENT!

  20. Congruence Postulates and Theorems • SSS • SAS • ASA • AAS • AAA? • SSA? • HL

  21. Name That Postulate (when possible) SAS ASA SSA AAS Not enough info!

  22. Name That Postulate (when possible) AAA Not enough info! SSS SSA SSA Not enough info! HL

  23. Name That Postulate (when possible) Not enough info! Not enough info! SSA SSA HL AAA Not enough info!

  24. Vertical Angles, Reflexive Sides and Angles When two triangles touch, there may be additional congruent parts. Vertical Angles Reflexive Side side shared by two triangles

  25. Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA AAS Not enough info!

  26. Reflexive Sides and Angles When two triangles overlap, there may be additional congruent parts. Reflexive Side side shared by two triangles Reflexive Angle angle shared by two triangles

  27. Let’s Practice ACFE Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AF For AAS:

  28. What’s Next Try Some Proofs End Slide Show

  29. End Slide Show Choose a Problem. Problem #1 SSS Problem #2 SAS Problem #3 ASA

  30. Problem #4 AAS Given Vertical Angles Thm Given AAS Postulate

  31. Problem #5 HL Given ABC, ADC right s, Prove: Given 1. ABC, ADC right s Given Reflexive Property HL Postulate

  32. Congruence Proofs 1. Mark the Given. 2. Mark … Reflexive Sides or Angles/Vertical Angles Also: mark info implied by given info. 3. Choose a Method. (SSS , SAS, ASA,AAS ,HL) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

  33. segments angles segments angles angles Given implies Congruent Parts midpoint parallel segment bisector angle bisector perpendicular

  34. Example Problem

  35. … and what it implies Step 1: Mark the Given

  36. Reflexive Sides • Vertical Angles Step 2: Mark . . . … if they exist.

  37. Step 3: Choose a Method SSS SAS ASA AAS HL

  38. STATEMENTS REASONS S A S Step 4: List the Parts … in the order of the Method

  39. STATEMENTS REASONS Step 5: Fill in the Reasons S A S (Why did you mark those parts?)

  40. STATEMENTS REASONS 1. 2. 3. 4. 5. 1. 2. 3. 4. 5. Step 6: Is there more? S A S

  41. Congruent Triangles Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides/Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more?

  42. Using CPCTC in Proofs • According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent. • This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles. • This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

  43. Corresponding Parts of Congruent Triangles • For example, can you prove that sides AD and BC are congruent in the figure at right? • The sides will be congruent if triangle ADM is congruent to triangle BCM. • Angles A and B are congruent because they are marked. • Sides MA and MB are congruent because they are marked. • Angles 1 and 2 are congruent because they are vertical angles. • So triangle ADM is congruent to triangle BCM by ASA. • This meanssides AD and BC are congruent by CPCTC.

  44. Corresponding Parts of Congruent Triangles • A two column proof that sides AD and BC are congruent in the figure at right is shown below:

  45. Corresponding Parts of Congruent Triangles • A two column proof that sides AD and BC are congruent in the figure at right is shown below:

  46. Corresponding Parts of Congruent Triangles • Sometimes it is necessary to add an auxiliary line in order to complete a proof • For example, to prove ÐR @ ÐO in this picture

  47. Corresponding Parts of Congruent Triangles • Sometimes it is necessary to add an auxiliary line in order to complete a proof • For example, to prove ÐR @ ÐO in this picture

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