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CONGRUENT TRIANGLES

CONGRUENT TRIANGLES. When we talk about congruent triangles, we mean everything about them is congruent (or exactly the same) All 3 pairs of corresponding angles are equal…. and all 3 pairs of corresponding sides are equal. . . . .

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CONGRUENT TRIANGLES

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  1. CONGRUENT TRIANGLES

  2. When we talk about congruent triangles, we mean everything about them is congruent (or exactly the same) All 3 pairs of corresponding angles are equal…. and all 3 pairs of corresponding sides are equal    

  3. For us to prove that 2 people are identical twins, we don’t need to show that all “2000 +” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 4 short cuts or rules to help us prove that two triangles are congruent.

  4. SSS If 3 corresponding sides are exactly the same length, then the triangles have to be congruent. Side-Side-Side (SSS)

  5. SAS Non-included angles Included angle If 2 pairs of sides and the included angles are exactly the same then the triangles are congruent. Side-Angle-Side (SAS)

  6. AAS If 2 angles and a side are exactly the same then the triangles have to be congruent. Angle-Angle-Side (AAS)  

  7. RHS If 2 triangles have a right-angle, a hypotenuse, and a side, which are exactly the same then the triangles are congruent. Right angle-Hypotenuse-Side (RHS)

  8. This is called a common side. It is a side for both triangles. Sometimes it is shown by a ‘squiggly’ line placed on the common line

  9. Why is AAA (Angle, Angle, Angle) NOT a proof for CONGRUENT TRIANGLES??? Discuss!!

  10. Which method can be used to prove the triangles are congruent b) a) d) c)

  11. 3Common Sides (SSS) Included angle, 2 sides equal (SAS) RHS Right angle Hypotenuse & a side equal Parallel lines alternate angles Common side (AAS)

  12. RHS is used only with right-angled triangles, BUT, not all right triangles. ASA RHS

  13. PART 2

  14. When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Information to do this

  15. A C B E D Given: AB = BD EB = BC Aim: Prove ∆ABE  ∆DBC

  16. A C Given: AB = BD EB = BC Mark these on the diagram Aim: Prove ∆ABE ∆DBC B E D Proof: AB = BD (Given) SIDE (S) ABC = CBD (Vertically Opposite) ANGLE (A) EB = BC (Given) SIDE (S)  ∆ABE  ∆DBC (SAS)

  17. C A B X Given:  CX bisects  ACB CAB = CBA Prove: ∆ACX ∆BCX

  18. C   A B X Proof:  ACX =  BCX CX bisects  ACB (ANGLE) (A) CAB = CBA (Given) ANGLE (A) CX is common SIDE (S)  ∆ACX ∆BCX (AAS)

  19. Given: MN ║ QR MN = QR Prove: ∆MNP ∆QRP N M P R Q

  20. N M  P Proof:  MNP =  QRP Alternate angles MN║QR (ANGLE) (A) MPN = QPR Vertically Opposite angles (ANGLE) (A) MN = QR Given (SIDE) (S)  ∆MNP ∆QRP(AAS) R Q

  21. C Y X Given: XZ = AC XY = AB  XYZ= ABC = 90 Prove: ∆ABC ∆XYZ B A Z

  22. C Y X A B Z Proof:  ABC =  XYZ = 90Given (RIGHT-ANGLE) (R) XZ = AC Given (Hypotenuse) (H) XY = AB Given (SIDE) (S) ∆XYZ ∆ABC (RHS)

  23. P Q Given: PQ = RS QR = SP Prove: ∆PQR ∆RSP R S

  24. P Q R S Proof: PQ = RS Given (SIDE) (S) QR = SP Given (SIDE) (S) XY = ABCommon to both (SIDE) (S) ∆PQR ∆RSP (SSS)

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