9
The second test for similar triangles
Textbook: pp. 34–35
GoalKnow and apply the second similarity test for triangles (SAS).
New words
SAS test (side-angle-side) · TBT alomatiincluded angle · tomonlar orasidagi burchakvertical angles · vertikal burchaklarratio of areas · yuzlar nisbati
Explanation
Second test (SAS): if two sides of one triangle are proportional to two sides of another and the angles between those sides are equal, the triangles are similar. The angle must lie between the given sides. Often the equal angle is common to both triangles or is a pair of vertical angles. Once similarity is known, the same coefficient applies to the third side, and the ratio of areas is k². Do not confuse this with the SAS test for congruence: here the sides are proportional, not equal.
Worked examples
Segments AB and CD meet at O with OA = 9, OB = 6, OC = 12, OD = 8. 9/6 = 12/8 = 3/2 and ∠AOC = ∠BOD (vertical). So △AOC ∽ △BOD with k = 3/2 and area ratio 9/4.
In △ABC and △A₁B₁C₁, ∠A = ∠A₁, AB = 4, AC = 6, A₁B₁ = 6, A₁C₁ = 9: 6/4 = 9/6 = 3/2, so they are similar. If BC = 5, then B₁C₁ = 7.5.
Class activity
“Find the match”: the teacher hands out cards with sides and angles; students find pairs of triangles similar by SAS and state k.
Practice
1
In △ABC and △DEF, ∠B = ∠E, AB = 6, BC = 10, DE = 9, EF = 15. Are the triangles similar? Find k.
Yes, since 9/6 = 15/10 = 3/2; k = 3/2.
2
△AOC ∽ △BOD with k = 2 and BD = 7 cm. How long is AC (cm)?
14
3
What is the ratio of areas of similar triangles when k = 3?
9
4
Why must the angle lie between the two given sides?
Otherwise one can draw two triangles with the same sides and angle but different shapes.