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Homework Proof Problem #18

4.1 Detours and Midpoints. Homework Proof Problem #18. 18) Given: AD  DB AE  BC CD  ED Prove: Δ AFB is isosceles. A. B. F. E. C. D. 18) Given: AD  DB AE  BC CD  ED Prove: Δ AFB is isosceles. A. B. F. E. C. D. From our givens: Δ AED  Δ BCD

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Homework Proof Problem #18

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  1. 4.1 Detours and Midpoints Homework ProofProblem #18

  2. 18) Given: AD  DB AE  BC CD  ED Prove: Δ AFB is isosceles A B F E C D

  3. 18) Given: AD  DB AE  BC CD  ED Prove: Δ AFB is isosceles A B F E C D

  4. From our givens: ΔAED ΔBCD by : SSS A B Now we have 3 pairs of corresponding congruent angles! E C D For our next pair of Δ’s we will need: ∡EDA  ∡CDB by CPCTC

  5. We have: ED  DC AD  DB and PART of the included angle between those two sides! A B 2 pairs of congruent sides We need the rest of ∡EDB and ∡CDA E C ∡ADB  ∡ADB by REFLEXIVE PROPERTY That give us . . . ∡EDB  ∡CDA by ADDITION D From these triangles, we get sides EB  AC by CPCTC ΔEDB ΔCDA by SAS

  6. We are getting closer to ΔAFB! A B For this pair of Δ’s, we already have: AE  BC (Given) EB  CA (CPCTC) F This means: ∡ABE  ∡BAC by CPCTC E C We have another EASY side! AB  AB by Reflexive Property Now we have proven: ΔEABΔCBA by SSS Look what we have! 

  7. A B It’s time for: F Theorem 21: (Angle-Side Theorem) If two angles of a triangle are congruent, the sides opposite the angles are congruent. So now we can say: AF  BF Reason? IF , then And for our final “PROVE!” Statement: Δ AFB is ISOSCELES Reason:If a triangle has at least two congruent sides, then it is isosceles byDEFINITION!

  8. Statements Reasons A B 1. AD  DB (S) 1. Given 2. AE  BC (S) 2. Given 3. CD  ED (S) 3. Given F 4. Δ AED  ΔBCD 4. SSS (1, 2, 3) 5. ∡ EDA  ∡CDB 5. CPCTC E C 6. ∡ ADB  ∡ADB 6. Reflexive Property 7. ∡EDB ∡CDA (A) 7. AdditionProperty D 8. Δ EDB  ΔCDA 8. SAS (1, 7, 3) 9. EB  CA 9. CPCTC 10. AB  AB 10. Reflexive Property 11. Δ EAB  ΔCBA 11. SSS (2, 10, 9) 12. ∡ ABE  ∡BAC 12. CPCTC 13. BF  AF 13. If , then 14. Δ AFB is isosceles 14. If a Δ has at least 2  sides, then isosceles!

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