Proving Triangles Congruent - Interactive Activity

Two triangles are congruent if and only if their corresponding parts are congruent — all three pairs of corresponding sides and all three pairs of corresponding angles. Proving congruence does not require showing that all six pairs of parts match. Five shortcuts — SSS, SAS, ASA, AAS, and HL — each name a minimum set of congruent parts that is enough to guarantee that the two triangles are congruent. The name of each shortcut is critical: it tells you exactly which parts you need and where those parts must sit in the picture.

The Five Shortcuts

SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Congruent sides are equal in length.
SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. The included angle is the angle formed by two given sides — the angle that lies between them, sharing their common vertex. Congruent angles are equal in measure.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The included side is the side between the two angles — its endpoints are the vertices of those two angles.
AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. For AAS, the side must sit in the corresponding position in both triangles.
HL (Hypotenuse-Leg): If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. HL works only for right triangles. The hypotenuse is the side opposite the right angle, and the legs are the two sides that form the right angle.

The working strategy: process of elimination

Mark the congruent parts in the picture, name the corresponding parts, and then test the marked parts against the list of shortcuts, eliminating each shortcut the marked parts do not satisfy. Free information often completes a shortcut: a shared side is congruent to itself (the reflexive property), vertical angles — the opposite angles formed when two lines intersect — are congruent (equal in measure), and a midpoint divides a segment into two congruent segments (equal in length). Once congruence is proved, write the congruence statement with the vertices in corresponding order and name the shortcut used.

Warning — two combinations never work. SSA (two sides and a non-included angle) and AAA (three angles) are not valid shortcuts. Neither one determines one unique triangle, so neither one can prove that two triangles are congruent. Three congruent angles only show that two triangles have the same shape; they say nothing about size.

The Congruence Explorer

What to do: Press a shortcut button. The triangles are re-marked each time so that the marks show exactly the parts that shortcut requires — matching tick marks show sides that are congruent (equal in length), and matching angle arcs show angles that are congruent (equal in measure). A small square marks a right angle. For HL, the activity shows a pair of right triangles, because HL applies only to right triangles.

What to notice: the marks move when the shortcut changes. Watch where the marked angle sits between the marked sides for SAS, where the marked side sits between the marked angles for ASA, and where the hypotenuse and the leg sit for HL.

Included or Not?

What to do: Press the name of a side or an angle of triangle ABC. The diagram highlights your choice together with the parts it belongs with, and the readout names those parts.

What to notice: a side is named by its two endpoints, and those endpoints are exactly the vertices of the two angles it is included between. An angle is formed by the two sides that share its vertex, and it is the included angle for exactly those two sides.

Name the Shortcut

What to do: Look at the marks on the two triangles — matching tick marks for congruent sides (equal in length), matching angle arcs for congruent angles (equal in measure), and a small square for a right angle. Then choose the one shortcut that the marked parts satisfy, or choose Not Congruent when no shortcut fits. If your first answer is wrong, you will get a hint and another try; after a second miss the explanation appears. Press New Challenge for a new diagram.

Which shortcut proves these two triangles congruent?

Your Turn

Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.