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Congruent triangles theorem

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Congruent triangles theorem

  1. 1. Geometry Triangle Congruence Theorems
  2. 2. Congruent triangles have three congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles. Congruent Triangles
  3. 3. The Triangle Congruence Postulates &Theorems LAHALLHL FOR RIGHT TRIANGLES ONLY AASASASASSSS FOR ALL TRIANGLES
  4. 4. Theorem If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. Think about it… they have to add up to 180°.
  5. 5. A closer look... If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. But do the two triangles have to be congruent? 85° 30° 85° 30°
  6. 6. Example 30° 30° Why aren’t these triangles congruent? What do we call these triangles?
  7. 7. So, how do we prove that two triangles really are congruent?
  8. 8. ASA (Angle, Side, Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . then the 2 triangles are CONGRUENT! F E D A C B
  9. 9. AAS (Angle, Angle, Side) Special case of ASA If two angles and a non- included side of one triangle are congruent to two angles and the corresponding non- included side of another triangle, . . . then the 2 triangles are CONGRUENT! F E D A C B
  10. 10. SAS (Side, Angle, Side) If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . then the 2 triangles are CONGRUENT! F E D A C B
  11. 11. SSS (Side, Side, Side) In two triangles, if 3 sides of one are congruent to three sides of the other, . . . F E D A C B then the 2 triangles are CONGRUENT!
  12. 12. HL (Hypotenuse, Leg) If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B F E D then the 2 triangles are CONGRUENT!
  13. 13. HA (Hypotenuse, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! F E D A C B
  14. 14. LA (Leg, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B F E D
  15. 15. LL (Leg, Leg) If both pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B F E D
  16. 16. Example 1 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? F E D A C B
  17. 17. Example 2 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B F E D
  18. 18. Example 3 Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D A C B
  19. 19. Example 4 Why are the two triangles congruent? What are the corresponding vertices? A B C D E F SAS A   D C   E B   F
  20. 20. Example 5 Why are the two triangles congruent? What are the corresponding vertices? A B C D SSS A   C ADB   CDB ABD   CBD
  21. 21. Example 6 Given: B C D A CDAB ADBC CDAB DABC CAAC Are the triangles congruent? S S S Why?
  22. 22. Example 7 Given: QRPS  R H SRSSR  Are the Triangles Congruent? QSR  PRS = 90 Q RS P T mQSR = mPRS = 90 PSQR  Why?
  23. 23. Summary: ASA - Pairs of congruent sides contained between two congruent angles SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides AAS – Pairs of congruent angles and the side not contained between them.
  24. 24. Summary --- for Right Triangles Only: HL – Pair of sides including the Hypotenuse and one Leg HA – Pair of hypotenuses and one acute angle LL – Both pair of legs LA – One pair of legs and one pair of acute angles
  25. 25. THE END!!!

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