24
Second criterion of congruence (ASA)
Textbook: pp. 62–63
GoalKnow and apply the ASA criterion.
New words
ASA criterion · BTB alomatiadjacent angles of a side · yopishgan burchaklarto prove · isbotlashcorresponding angles · mos burchaklar
Explanation
Second criterion (ASA): if one side and the two angles adjacent to it in one triangle are equal to one side and the two angles adjacent to it in another triangle, the triangles are equal. Idea of the proof: place the triangles so that the equal sides coincide and the third vertices lie on one side of it. Because the angles are equal, the other sides lie along the same rays, so the third vertices coincide and the triangles overlap exactly. In problems, once triangles are shown equal, we conclude that corresponding sides and angles are equal.
Worked examples
In △ABC and △A₁B₁C₁, AB = A₁B₁ = 7, ∠A = ∠A₁ = 50°, ∠B = ∠B₁ = 60° — by ASA △ABC = △A₁B₁C₁.
Segments AC and BD meet at O, BO = OD, ∠OBA = ∠ODC. Vertical angles ∠AOB = ∠COD, so by ASA △AOB = △COD and AB = CD.
Class activity
“Measuring a width”: a puzzle using ASA — draw two imaginary equal triangles and measure the bank side. Do not go near water; work on paper.
Practice
1
In △ABC and △DEF, AB = DE = 8, ∠A = ∠D, ∠B = ∠E. If AC = 11, find DF.
11
2
In ASA, how are the side and the angles placed?
The angles lie at the ends of the side.
3
With BO = OD and ∠OBA = ∠ODC, which third equality is needed to prove △AOB = △COD?
∠AOB = ∠COD (vertical angles)
4
Why must the angles in ASA be adjacent to the side?
Otherwise the third vertex is not uniquely determined.