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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 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. Segment QR is congruent to segment TV because QR = TV = 5. Triangle PQR is congruent to triangle STV by Side-Angle-Side. Back to Top  Side-Side-Side is a rule used in geometry to prove triangles congruent.  The rule states that if three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent. 1. Problem: Show that triangle QYN is congruent to triangle QYP. 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. Back to Top  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. Back to Top  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. Back to Top  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. Back to Top  Take the Quiz on congruent triangles.  (Very useful to review or to see if you've really got this topic down.)  Do it!