Triangle Congruence
Lessons in this chapter
About Triangle Congruence
Triangle Congruence teaches the shortcut theorems that prove two triangles congruent from just three carefully chosen measurements.
Congruent Polygons defines what it means for two figures to be exactly identical in size and shape, before SAS and SSS Congruence Theorems introduce the first two shortcuts, matching two sides and the included angle, or matching all three sides.
Equilateral and Isosceles Triangles covers the special angle relationships in triangles with two or three equal sides, and HL, ASA, and AAS Congruence Theorems close the chapter with the remaining shortcuts for proving triangles congruent, including the special case for right triangles.
Each of the four lessons here opens as a Geometry worksheet in the page: Congruent Polygons, SAS and SSS Congruence Theorems, Equilateral and Isosceles Triangles, and HL, ASA, and AAS Congruence Theorems. Every question is scored as you answer it, which is where most of the value is in this chapter, since choosing the wrong criterion is usually about which parts are given rather than about the triangles themselves. The practice is free to try, with no login needed for your first 3.
Frequently asked questions
Congruent Polygons sets out what congruence means for a polygon and why the order of the letters in a congruence statement matters. SAS and SSS Congruence Theorems and HL, ASA, and AAS Congruence Theorems then give the short lists of parts that are enough to guarantee two triangles are congruent, so a full check of all six parts is never needed. Equilateral and Isosceles Triangles covers the base angle results that follow from congruence. The chapter sits in the high school Geometry conceptual category (G).
SAS (side-angle-side) means if two sides and the INCLUDED angle (the angle between them) of one triangle match two sides and the included angle of another, the triangles are congruent.
SSS (side-side-side) means if all three sides of one triangle match all three sides of another, the triangles are congruent, regardless of any angle measurements.
ASA (angle-side-angle) requires the side to be BETWEEN the two given angles; AAS (angle-angle-side) allows the side to be adjacent to just one of the two angles instead: both are valid, but they describe different arrangements of the given information.
HL (hypotenuse-leg) proves two right triangles congruent from just their hypotenuse and one leg. It only works for right triangles because the right angle guarantees the missing leg's length is fixed by the Pythagorean Theorem, which isn't true for a general triangle.
AAA only proves triangles similar (same shape, not necessarily same size), and SSA can genuinely produce two different, non-congruent triangles from the same three measurements: neither guarantees one unique congruent triangle the way SAS, SSS, ASA, AAS, and HL do.