Proving Δ s are  : SSS, SAS, HL, ASA, & AAS

1. Proving Δs are  : SSS, SAS, HL, ASA, & AAS

2. Postulate (SSS)Side-Side-Side  Postulate • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

3. E A F C D B More on the SSS Postulate If seg AB  seg ED, seg AC  seg EF, & seg BC  seg DF, then ΔABC ΔEDF.

4. Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT. Example 3: S Q R T

5. DFGHJK SideDG HK, SideDF JH,andSideFG JK. So by the SSS Congruence postulate, DFG HJK. for Example 1 GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Yes. The statement is true.

6. 3. QPTRST GIVEN : QT TR , PQ SR, PT TS PROVE : QPTRST It is given that QT TR, PQ SR, PT TS.So by SSS congruence postulate, QPT RST. Yes the statement is true. PROOF: for Example 1 GUIDED PRACTICE Decide whether the congruence statement is true. Explain your reasoning. SOLUTION

7. Postulate (SAS)Side-Angle-Side  Postulate • If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

8. More on the SAS Postulate • If seg BC  seg YX, seg AC  seg ZX, & C X, then ΔABC  ΔZYX. B Y ) ( A C X Z

9. Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG. Example 4: D R A G

10. Theorem (HL)Hypotenuse - Leg  Theorem • If the hypotenuse and a leg of a right Δ are  to the hypotenuse and a leg of a second Δ, then the 2 Δs are . Note: Right Triangles Only

11. Postulate (ASA):Angle-Side-Angle Congruence Postulate • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

12. Theorem (AAS): Angle-Angle-Side Congruence Theorem • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

13. Proof of the Angle-Angle-Side (AAS) Congruence Theorem Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF D A B F C Paragraph Proof You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF. E

14. Theroem (CPCTC)Corresponding Parts of Congruent Triangles are Congruent When two triangles are congruent, there are 6 facts that are true about the triangles: • the triangles have 3 sets of congruent (of equal length) sides and  • the triangles have 3 sets of congruent (of equal measure) angles. Use this after you have shown that two figures are congruent. Then you could say that Corresponding parts of the two congruent figures are also congruent to each other.

15. Example 5: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

16. What Theorems were not used? • AAA • SSA Don’t make an “Angle Side Side” of yourself!

17. Example 5: In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

18. Example 6: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.