Congruent Triangles Featuring SSS and SAS (side-side-side and side-angle-side)

Congruent Triangles  We know that in order for polygons to be congruent all parts of the first polygon must be congruent with all the corresponding parts of the second polygon.  All sides must be congruent  All angles must be congruent  For triangles we have three sides and three angles, so there are six items that have to be congruent.  Remember: the order of the letters used for sides and angles is extremely important! Each side and angle designation MUST match the corresponding side or angle.

Congruent Triangles  The example below note how the letters are ordered in the congruency relation. ABCD ≅ EFGH

Congruent Triangles  If the polygons to the right are congruent (HIJK ≅ LMNO), what are the corresponding parts that are congruent?  Sides:  Angles:

Congruent Triangles  Figures (drawings of polygons) can be translated, rotated and reflected without losing congruency. That means that two polygons might look different because they have been flipped over, turned around or moved. However, as long as the corresponding parts are congruent, the polygons are still congruent!  We can also use these transformations to prove that two figures are congruent by moving one object until it overlays (matches) the other polygon.

Your turn!  Work with a partner to determine whether triangles to the right are congruent. Does it matter if we break apart the triangles?  Answer: triangle ABC ≅ EDC

The Third Angle Theorem  If two angles of a triangle are congruent with two angles from a second triangle then the third angle in each triangle must be congruent. (Note this can be proven in a proof very much like the one we used to show that three angles of a triangle must add up to 180 degrees.)

Proofs with congruent triangles Let’s say that we wanted to prove that two triangles are congruent by using a formal proof. Here’s one way in which we might do that.