Triangle Congruence Theorems

Triangle Congruence Theorems

On this page, we hope to clear up problems that you might have with proving triangles congruent.  Triangles are one of the most used figures in geometry and beyond (engineering), so they are rather important to understand.  Scroll down or click any of the links below to start understanding congruent triangles better!

Side-Angle-Side
Side-Side-Side
Angle-Side-Angle
Angle-Angle-Side
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Quiz on Congruent Triangles

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Side-Angle-Side is a rule used in geometry to prove triangles congruent.  The rule states that if two sides and the included angle are congruent to two sides and the included angle of a second triangle, the two triangles are congruent.  An included angle is an angle created by two sides of a triangle.

1. Problem: Is triangle PQR congruent to
            triangle STV by SAS? Explain.

  
  Solution: Segment PQ is congruent
              to segment ST because
              PQ = ST = 4.
            Angle Q is congruent to
              angle T because
              angle Q = angle T = 100 degrees.

 Solution: Segment QN is congruent to
              segment QP and segment YN is
              congruent to segment YP because that
              information is given in the figure.
            Segment YQ is congruent to segment
              YQ by the Reflexive Property of Con-
              gruence, which says any figure is 
              congruent to itself.
            Triangle QYN is congruent to triangle
              QYP by Side-Side-Side.
  

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Angle-Side-Angle is a rule used in geometry to prove triangles are congruent.  The rule states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.  An included side is a side that is common to (between) two angles.  For example, in the figure used in the problem below, segment AB is an included side to angles A and B.

1.   Problem: Show that triangle BAP is congruent to triangle CDP.



Solution: Angle A is congruent to angle D because they are both right angles. Segment AP is congruent to segment DP be- cause both have measures of 5. Angle BPA and angle CPD are congruent be-

cause vertical angles are congruent.
               Triangle BAP is congruent to triangle CDP
                  by Angle-Side-Angle.

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Angle-Angle-Side is a rule used in geometry to prove triangles are congruent.  The rule states that if two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, the two triangles are congruent.

1.   Problem: Show that triangle CAB is congruent
              to triangle ZXY.







     Solution: Angle A and angle Y are congruent
                  because that information is given 
                  in the figure.
               Angle C is congruent to angle Z 
                  because that information is given 
                  in the figure.
               Segment AB corresponds to segment XY and
                  they are congruent because that
                  information is given in the figure.
               Triangle CAB is congruent to triangle ZXY
                  by Angle-Angle-Side.

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When two triangles are congruent, all six pairs of corresponding parts (angles and sides) are congruent.  This statement is usually simplified as corresponding parts of congruent triangles are congruent, or CPCTC for short. 

1.   Problem: Prove segment BC is congruent to
              segment CE.
                
  


  
  


     Solution: First, you have to prove that triangle
               CAB is congruent to triangle CED.
               
               Angle A is congruent to angle D 
                  because that information is given 
                  in the figure.
               Segment AC is congruent to segment CD
                  because that information is given
                  in the figure.
               Angle BCA is congruent to angle DCE
                  because vertical angles are
                  congruent.
               Triangle CAB is congruent to triangle CED
                  by Angle-Side-Angle.
               
               
               Now that you know the triangles are 
               congruent, you know that all 
               corresponding parts must be congruent.  
               By CPCTC, segment BC 

               is congruent to segment CE