Geometry: Congruent Triangles

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. Read on, or follow 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

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.

Example:

1. Problem:  Is triangle PQR in the accompanying figure 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.

Side-Side-Side

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 triagnles are congruent.

Example:

1. Problem:  Show that triangle QYN in the accompanying figure 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 Congruence, which says any figure is
             congruent to itself.
             Triangle QYN is congruent to triangle QYP by
             Side-Side-Side.

Angle-Side-Angle

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.

Example:

1. Problem:  Show that triangle BAP in the accompanying figure 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 because both
             have a measure of 5.
             Angle BPA and angle CDP are congruent because
             vertical angles are congruent.
             Triangle BAP is congruent to triangle CDP by
             Angle-Side-Angle.

Angle-Angle-Side

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.

Example:

1.  Problem: Show that triangle CAB in the accompanying figure 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.

CPCTC

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.

Example:

1.  Problem: Prove segment BC in the accompanying figure 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.

Take the quiz on congruent triangles. The quiz is very useful for either review or to see if you've really got the topic down.

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