22
First criterion of congruence (SAS)
Textbook: pp. 58–59
GoalKnow equal triangles and the SAS criterion.
New words
equal (congruent) triangles · teng uchburchaklarSAS criterion · TBT alomaticorresponding sides · mos tomonlartwo sides and the included angle · ikki tomon va ular orasidagi burchak
Explanation
Triangles that coincide exactly when one is laid on the other are called equal (congruent); writing △ABC = △A₁B₁C₁ means the vertices correspond in order, so corresponding sides and angles are equal. First criterion (SAS): if two sides and the angle between them in one triangle are equal to two sides and the angle between them in another, the triangles are equal. Idea of the proof: lay the equal angles on each other; since the equal sides lie on the rays, the endpoints also coincide. In figures, equal sides and angles are marked with the same signs.
Worked examples
In △ABC and △A₁B₁C₁, AB = A₁B₁ = 5, AC = A₁C₁ = 8, ∠A = ∠A₁ = 40° — by SAS the triangles are equal, so BC = B₁C₁.
Segments AC and BD cross at O and bisect each other (AO = OC, BO = OD). Since vertical angles ∠AOB = ∠COD are equal, △AOB = △COD (SAS), hence AB = CD.
Class activity
“Find the pair”: cards give two sides and an angle of triangles; pupils find the pairs equal by SAS.
Practice
1
In △ABC and △DEF, AB = DE = 6, AC = DF = 9, ∠A = ∠D. If EF = 7, find BC.
7
2
What can we say about corresponding angles of equal triangles?
They are equal.
3
O is the midpoint of both AC and BD. By which criterion do you prove △AOB = △COD?
SAS: AO = OC, BO = OD, vertical angles equal.
4
Why must the angle in SAS be the one between the two sides?
Otherwise two sides and another angle do not determine the triangle uniquely.