25
Third criterion of congruence (SSS)
Textbook: pp. 64–65
GoalKnow SSS and explain the rigidity of a triangle.
New words
SSS criterion · TTT alomatirigid shape · mustahkam shaklcorollary · natijacorresponding angles · mos burchaklar
Explanation
Third criterion (SSS): if the three sides of one triangle are equal to the three sides of another, the triangles are equal. Idea of the proof: put the equal largest sides together and join the third vertices; the two isosceles triangles formed have equal base angles, so the opposite angles are equal and SAS applies. Corollary: triangles with three pairs of equal sides have equal corresponding angles. That is why the triangle is a rigid shape: with fixed side lengths its angles cannot change, while a quadrilateral can fold. This is used in bridges and roof frames.
Worked examples
In △ABC and △MNK with AB = MN = 5, BC = NK = 6, AC = MK = 7, SSS gives equal triangles; hence ∠A = ∠M.
Two strips joined by a nail can change their angle; adding a third strip makes a triangle and the shape locks.
Class activity
“Rigid or not?”: make a quadrilateral and a triangle from cardboard strips (an adult makes the holes) and test which one folds.
Practice
1
In △ABC and △DEF, AB = DE, BC = EF, AC = DF. If ∠B = 70°, find ∠E (degrees).
70
2
How many pairs of equal sides are needed for SSS?
3
3
How is a four-sided door frame made rigid?
A diagonal brace is added, making two triangles.
4
Why is a triangle a rigid shape?
With given sides the triangle is determined by SSS — its angles cannot change.