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Warm Up

Warm Up. Given the line y = 2x + 10 , and given the point (- 4, 5). Write the equation of the line that is perpendicular to the given line and goes through the given point. Write the equation of the line that is parallel. Congruent Triangles 4.1-4.2. Today’s Goals:

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Warm Up

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  1. Warm Up Given the line y = 2x + 10, and given the point (- 4, 5). • Write the equation of the line that is perpendicular to the given line and goes through the given point. • Write the equation of the line that is parallel.

  2. Congruent Triangles4.1-4.2 Today’s Goals: To recognize congruent figures. To prove two triangles congruent using SSS and SAS.

  3. Proving Triangles Congruent Powerpoint hosted on www.worldofteaching.com Please visit for 100’s more free powerpoints

  4. Warm Up

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

  6. Congruent Polygons • If two polygons are congruent, then all the angles are congruent • And all the sides are congruent.

  7. F R D C J T Ex.1: Naming Congruent PartsTJD  RCF. List the congruent corresponding parts. • Sides:TJ  RC JD  CF DT  FR • Angles:T R J C D F

  8. Third Angles Theorem • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.  C   F

  9. PQ  PS, QR  SR PR  PR Q S, QPR SPR QRP SRP PQR  PSR Ex.2: Proving Triangles Congruent Use the information given in the diagram. Give a reason why each statement is true. Given Reflexive Property Given 3rd Angles Thm. Definition of Congruent Triangles

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

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

  12. The Short Cuts SSS SAS ASA AAS Do you need all six ? Is by definition the only way? NO !

  13. Rigid • If you have the three sides then there is no choice for the angles. • The triangle is rigid. • Remember how the 4,5,8 triangle worked yesterday?

  14. Side-Side-Side (SSS) E B F A D C • AB DE • BC EF • AC DF ABC DEF

  15. B A C Included Angle • The angle between two sides (segments). • B is included between AB and CB.

  16. Included Angle The angle between two sides H G I

  17. E Y S Included Angle Name the included angle: YE and ES ES and YS YS and YE E S Y

  18. Rigid? • When you hold a firm angle with your hands is the distance between your fingertips fixed?

  19. Pasta and Protractor • I taped pasta of lengths 5 inches and 3 inches on the protractor at a 25 degree angle. Hold the red pipe cleaner up for the third side. Do you have a choice as to how long the pipe cleaner can be?

  20. Side-Angle-Side (SAS) B E F A C D • AB DE • A D • AC DF ABC DEF included angle

  21. Included Side The side between two angles GI GH HI

  22. E Y S Included Side Name the included side: Y and E E and S S and Y YE ES SY

  23. I need four volunteers to demonstrate opposite and adjacent.

  24. Rigid? • Do the lines have a fixed point of intersection? • Try the pasta with angles.

  25. Angle-Side-Angle (ASA) B E F A C D • A D • AB  DE • B E ABC DEF included side

  26. Angle-Angle-Side (AAS) B E F A C D • A D • B E • BC  EF ABC DEF Non-included side

  27. Just a short cut. • If you know two angles of a triangle you can find the third.(They always add up to 180) Name that theorem. • Thus, we are using ASA. This allows us to skip finding the other angle. • ASA and AAS are the same.

  28. Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

  29. Never ever say triangles are congruent by “donkey” forward or backwards!

  30. Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

  31. You must have a side to know the size!

  32. SSS correspondence • ASA correspondence • SAS correspondence • AAS correspondence • SSA correspondence • AAA correspondence The Congruence Postulates

  33. Name That Postulate (when possible) SAS ASA SSA SSS

  34. Name That Postulate (when possible) AAA ASA SSA SAS

  35. Sometimes the corresponding parts are not marked but you know they are congruent. What should you look for?

  36. Look for: • Common Parts (Reflexive Property) • Vertical Angles • Angles formed by Parallel lines • Angles formed by Perpendicular Lines • Linear Pairs • Substitution

  37. Give the Supporting Fact then Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS

  38. Warning! • These extra facts are not reasons for the triangles to be congruent!

  39. What are the reasons why two triangles can be congruent? • Given • Definition • SSS • SAS • ASA • AAS and one more we will learn tomorrow!

  40. Try to Name That Postulate (when possible)

  41. Name That Postulate (when possible)

  42. 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:

  43. Cool Down Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:

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