Triangle congruence criteria
Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent
What a learner can do afterwards
- State which congruence criterion (SSS, SAS, ASA, or RHS) applies to a given pair of triangles
- Use the notation △ABC ≅ △DEF and match corresponding sides and angles
- Determine whether given measurements produce a unique triangle, more than one, or none
1 · Read
Every triangle has three vertices, named with capital letters like A, B, and C. Each side gets the matching lowercase letter of the vertex across from it. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Two triangles are congruent when they are exactly the same size and shape, even if one is turned or flipped. In symbols that is △ABC ≅ △DEF. The letter order matters: A matches D, B matches E, C matches F, so side AB equals side DE.
Triangle ABC has sides 5 cm, 6 cm, and 7 cm. Triangle DEF has sides 5 cm, 6 cm, and 7 cm too, matched in the same order. All three pairs of sides match, so the triangles are congruent by SSS (side-side-side). You do not even need to check the angles.
Four checks prove congruence. SSS: all three sides match. SAS: two sides and the angle between them match. ASA: two angles and the side between them match. RHS: in a right triangle, the right angle, the hypotenuse, and one other side match. For SAS and ASA, the letter in the middle of the name must sit between the other two. RHS is different: its letters do not follow the between rule.
Three side lengths that close up give exactly one triangle, which is why SSS works. The triangle inequality must hold: each side must be shorter than the other two added together. If one length is longer than or equal to the other two added together, no triangle can be built at all. Equality gives a degenerate flat triangle.
Two sides with an angle that is not between them do not always fix a single triangle: sometimes two different triangles fit, sometimes one, sometimes none. That is why SSA is not a general criterion, though RHS is a special case that does give uniqueness. Three matching angles fix the shape but not the size, so the triangles can be different sizes.
Match corresponding vertices in order, then check SSS, SAS, ASA, or RHS to prove two triangles are congruent.
Three readings, nine practice questions and six flashcards were written for this triangle congruence topic. The videos are linked from their original sources.
2 · Watch
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Where this leads
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17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.