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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.

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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.

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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.