NAME                                                                       DATE                       PERIOD 
                            4-3              Study Guide and Intervention
                                             Congruent Triangles
                          Congruence and Corresponding Parts                                                                       B                          S
                          Triangles that have the same size and same shape are                                                               R
                          congruent triangles. Two triangles are congruent if and                                                                            T
                          only if all three pairs of corresponding angles are congruent                                        C               A
                          and all three pairs of corresponding sides are congruent. In 
                          the figure, △ABC  △RST.
                              Third Angles              If two angles of one triangle are congruent to two angles
                              Theorem                   of a second triangle, then the third angles of the triangles are congruent.
                              Example 
                           If △XYZ  △RST, name the pairs of                                                                           Y
                          congruent angles and congruent sides.                                                                       S
                          ∠X  ∠R, ∠Y  ∠S, ∠Z  ∠T                                                                X                                        R
                          −−       −− −−          −− −−          −−                                                             Z              T
                          XY     RS  ,  XZ     RT  ,  YZ     ST  
                           
                          Exercises
                          Show that the polygons are congruent by identifying all congruent corresponding 
                          parts. Then write a congruence statement.
                                                K                                                B                         K                     L
                           1.                                    2.                                             D  3. 
                                 B
                                              J               L              A
                               A                C                                            C                              J                    M
                            ∠A  ∠J; ∠B  ∠K;                                ∠A  ∠D; ∠ABC  ∠DCB  ∠J  ∠L; ∠JKM  ∠LMK;
                                               −− −−                                                  −− −−                                         −−       −−
                            ∠C  ∠L;   AB      JK ;                         ∠ACB  ∠DBC;   AC      BD                    ∠KMJ  ∠MKL;   KJ      ML  
                               −− −− −− −−                                   −− −−                                          −−       −−
                              BC      KL ;     AC      JL                    AB      DC                                    KL      MJ  
                            △ABC  △JKL  △ABC  △DCB  △JKM  △LMK                                                                                                        Lesson 4-3
                                                                                                                           
                                                                             B                D                            R
                           4.                                   5.                                   6. 
                                                LK
                               FG
                                                                                                                           US
                               E                             J                    A              C                          T
                                 
                            ∠E  ∠J; ∠F  ∠K;                                ∠A  ∠D;                                      ∠R  ∠T; 
                                               −−       −−
                            ∠G  ∠L;   EF      JK ;                         ∠ABC  ∠DCB;                                  ∠RSU  ∠TSU;
                               −− −− −− −−
                                 EG      JL ;     FG      KL ;             ∠ACB  ∠DBC;                                  ∠RUS  ∠TUS; 
                            
                                                                             −− −− −− −−                                   −− −− −− −−
                            △FGE  △KLJ    AB      DC ;    AC      DB ;                                                    RU      TU ;     RS      TS ; 
                                                                             −− −−                                         −− −−
                                                                              BC     CB  ; △ABC  △DCB    SU      SU ;  △RSU  △TSU
                                                                            
                Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.                                                       &
                          Suppose △ABC  △DEF                                                                        #
                                                                                                                                      % (2y-5)°         2x +y
                                                                                                                         °
                                                              27.8                                                     75
                           7. Find the value of x.                                                              64.3            90.6
                                                                                                                      °            °
                           8. Find the value of y.  35                                                             65            40
                                                                                                                "         96.6          $                 '
                          Chapter 4 19 Glencoe Geometry
                     NAME                                                                                                          DATE                                   PERIOD 
                       4-3                      Study Guide and Intervention  (continued)
                                                Congruent Triangles 
                     Prove Triangles Congruent Two triangles are congruent if and only if their 
                     corresponding parts are congruent. Corresponding parts include corresponding angles and 
                     corresponding sides. The phrase “if and only if” means that both the conditional and its 
                     converse are true. For triangles, we say, “Corresponding parts of congruent triangles are 
                     congruent,” or CPCTC.
                           Example                      Write a two-column proof.                                                                                                              $
                                         −− −−− −−−                         −−−
                                         AB    CB           ,    AD    CD         ,  ∠BAD  ∠BCD 
                      Given:    
                                         −−−                                                                                                               #                                     %
                                         BD       bisects ∠ABC.
                       Prove: △ABD  △CBD
                      Proof:                                                                                                                                                                  "
                       Statement                                                                             Reason
                            −− −−− −−−                         −−−                                           1. Given
                      1.  AB         CB       ,   AD        CD       
                            −−−          −−−                                                                 2. Reflexive Property of congruence
                      2.   
                            BD     BD           
                      3. ∠BAD  ∠BCD                                                                         3. Given
                      4. ∠ABD  ∠CBD                                                                         4. Definition of angle bisector
                      5. ∠BDA  ∠BDC                                                                         5. Third Angles Theorem
                      6. △ABD  △CBD                                                                         6. CPCTC
                     Exercises
                     Write a two-column proof.
                                                                                           −−−          −−− −−                −−−                                                                                                         Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
                      1. Given: ∠ A  ∠C, ∠D  ∠B,   
                                               −−                      −−−                 AD    CB          ,   AE       CE       ,    "
                                              AC   bisects  BD                .                                                                                                         #
                                                
                            Prove: △AED  △CEB                                                                                                                   &
                            Proof:                                                                                                        %                                             $
                            Statements                                                       Reasons
                            1. ∠A  ∠C, ∠D  ∠B                                              1. Given
                            2. ∠AED  ∠CEB                                                   2. Vertical angles are .
                                   −− −− −− −−                                               3. Given
                                  AD      CB ,   AE                   CE          
                            3.   
                                   −−            −−                                          4. Definition of segment bisector
                                  DE     BE              
                            4.   
                            5. △AED  △CEB                                                   5. CPCTC
                     Write a paragraph proof.                                                                                                                                                       #
                                             −−−
                      2. Given:    
                                             BD   bisects ∠ABC and ∠ADC,
                                             −− −−− −−                           −−− −−−                −−−
                                             AB         CB      ,   AB       AD      ,    CB     DC          
                            Prove: △ABD  △CBD
                                                                −−
                            We are given   BD  bisects                                        ∠ABC and ∠ADC. Therefore 
                            ∠ABD  ∠CBD and ∠ADB  ∠CDB by the definition                                                                                                         "                                    $
                            of angle bisectors. By the Third Angle Theorem, we 
                                                                                                                              −−             −− −− −−                                               %
                                                                                                                             AB     CB             ,    AB       AD         ,  
                            find that ∠A  ∠C. We are given that   
                                        −− −−
                                       CB     DC              . Using the substitution property, we can determine that  
                            and    −−
                             −−                                            −− −−
                            AD     CD             . Finally,              BD         BD          using the Reflexive Property of congruence.  
                            Therefore △ABD  △CBD by CPCTC.
                     Chapter 4 20 Glencoe Geometry