Name ___________________________________________________    Date  __________________
                           CCoongngrruueentnt T Trriiaangnglesles
                           Triangle Congruence Theorems
                           Five valid methods for proving that triangles are congruent are given below.
                                       SAS                            SSS                 HL (right triangles only)                            ASA                            AAS
                                                 E                              E                             E                                          E                              E
                                     B                             B                              B                                          B                              B
                                          D          F                  D           F                        D           F                        D           F                 D           F
                             A           C                  A           C                        A          C                         A           C                 A           C
                            Two sides and the                 All three sides              The hypotenuse and one                     Two angles and                 Two angles and a 
                            included angle                    are congruent.               of the legs are congruent.                 the included side              non-included side 
                            are congruent.                                                                                            are congruent.                 are congruent.
                           Example 1   Determine whether there is enough information to prove that the triangles are congruent. 
                                              Explain your reasoning.
                            a.    A                             B                                                    b.             K
                                                    C
                                 ED JML
                                                                                  — —
                            a.   You are given that ∠A ≅ ∠E and  AC    ≅  EC  . By the                      Vertical Angles Congruence Theorem, 
                                ∠ACB ≅ ∠ECD. So, two pairs of angles and their included sides are congruent. By the 
                                ASA Congruence Theorem, △ABC ≅ △EDC.
                                                            — —
                            b.   You are given that  JK    ≅  LK  .  You know that ∠J ≅ ∠L by the Base Angles Theorem. You also 
                                               — —
                                know that  KM    ≅  KM   by the Refl            exive Property of Segment Congruence. Because two pairs of 
                                sides and their non-included angles are congruent, you cannot conclude that △JKM ≅ △LKM.
                           Practice                                                                                                       Check your answers at BigIdeasMath.com.
                           Determine whether there is enough information to prove that the triangles are 
                           congruent. If so, state the theorem you would use.
                            1.                                                         2.                                                         3.  no
                                   XY EF
                                                                                                                                                                           R
                                               W                  Z
                                  yes; SSS Congruence Theorem                                D                                        G                  Q                 TS
                                                                                             yes; HL Congruence Theorem
                            4.      no                            C                    5.           G                             H               6.                                              U
                                   B
                                                                                                                                                         S                   T
                                                                                             FI V
                                                                   D                       yes; AAS Congruence Theorem
                                   A                                                                                                                     R
                                                                                                                                                        yes; ASA Congruence Theorem
                          Copyright © Big Ideas Learning, LLC                                                                                                                                   Topic 11.5