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CONGRUENT TRIANGLES

CONGRUENT TRIANGLES. When we talk about congruent triangles, we mean everything about them is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal. Corresponding parts

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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. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal

  3. Corresponding parts • When 2 figures are congruent the corresponding parts are congruent. (angles and sides) • If Δ ABC is  to Δ XYZ, which side is  to BC? YZ

  4. 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 5 short cuts or methods.

  5. SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.

  6. SAS Non-included angles Included angle Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent.

  7. ASA, AAS A ASA – 2 angles and the included side S A AAS – 2 angles and The non-included side A A S

  8. HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. ASA HL

  9. Some other stuff we’ll use…..

  10. This is called a common side. It is a side for both triangles. We’ll use the reflexive property To show it is congruent to itself.

  11. We’ll also use angle relationships we’ve learned to help us Vertical angles Alt int angles

  12. DEF OF MIDPOINT (will make sides congruent) • DEF OF A BISECTOR (will make sides or angles congruent) We’ll also use definition of bisector/midpoint

  13. Steps to show congruence 1. Add any congruent marks for common sides, vertical angles or alternate interior angles See if there is enough info to use SSS, SAS, AAS, ASA or HL 3. There is no ASS or AAA!

  14. Which method can be used to prove the triangles are congruent

  15. Common side SSS Vertical angles SAS Parallel lines alt int angles Common side SAS

  16. ExamplesIs it possible to prove the Δs are ? ( ) )) )) (( ) ( (( No, there is no AAA thm! Yes, ASA

  17. PROOFS Follow all the steps we havedone already but you will also need to add the GIVEN information onto the diagram

  18. Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC A C = B 1 2 SAS E D

  19. A C Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC B 1 2 = SAS E D STATEMENTS REASONS AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ABE ˜ ∆DBC SAS =

  20. C Given: CX bisects ACB A ˜ B Prove: ∆ACX˜ ∆BCX = 2 1 = AAS B A X CX bisects ACB Given 1 = 2 Def of angle bisc A = B Given CX = CX Reflexive Prop ∆ACX ˜ ∆BCX AAS =

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