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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-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.

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 triangles are congruent.

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.


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.


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.