The three-sides case (SSS) and the two-sides-and-included-angle case (SAS)
Mathematics Prep 3 Term 1, Unit 3, Lesson 2
GoalDecide similarity from three proportional sides, or from two proportional sides and the angle between them.
New words
SSS · all three pairs of sides are in the same ratioSAS · two pairs of sides are in the same ratio and the angles between them are equalincluded angle · the angle formed by the two given sidesorder the sides · pair the shortest with the shortest, and so on
Explanation
Sometimes we know sides, not angles. SSS: if the three sides of one triangle are proportional to the three sides of another, the triangles are similar. To test, order each triangle’s sides from the shortest to the longest and divide each pair; all three ratios must be equal. 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 be the included one. Two sides and an angle that is not between them (SSA) do not guarantee similarity. And one equal ratio of sides is not enough: all the required ratios must agree.
Case
What must be true
AA
two pairs of equal angles
SSS
three equal ratios of sides
SAS
two equal ratios and equal included angles
SSA
not a similarity case
Worked examples
Are the triangles with sides 4, 6, 8 and 6, 9, 12 similar? Order the sides: 4 ÷ 6 = 2/3, 6 ÷ 9 = 2/3, 8 ÷ 12 = 2/3. All three ratios equal: similar (SSS), k = 2/3. And 4, 6, 8 with 6, 9, 11? 4 ÷ 6 = 6 ÷ 9 = 2/3 but 8 ÷ 11 is not 2/3, so not similar.
△ABC has AB = 6, AC = 9, ∠A = 40°. △DEF has DE = 8, DF = 12, ∠D = 40°. Show they are similar and find BC when EF = 10. AB ÷ DE = 6/8 = 3/4 and AC ÷ DF = 9/12 = 3/4, and the included angles ∠A = ∠D = 40°. So △ABC ~ △DEF (SAS) with k = 3/4. Then BC = 10 × 3/4 = 7.5.
Two lines cross at E. AE = 6, EC = 9, BE = 4, ED = 6, CD = 12. Find AB. AE ÷ CE = 6/9 = 2/3 and BE ÷ DE = 4/6 = 2/3. The angles ∠AEB and ∠CED are vertical angles, so equal. △AEB ~ △CED (SAS), k = 2/3, and AB = 12 × 2/3 = 8.
Class activity
Straw triangles. Cut straws to make a triangle with sides 4, 6 and 8 units and another with 6, 9 and 12 units. Place one on top of the other with the same orientation. Do the angles match? Then try 6, 9 and 11.
Practice
1
Are the triangles with sides 5, 12, 13 and 10, 24, 26 similar? Give k.
5 ÷ 10 = 12 ÷ 24 = 13 ÷ 26 = 1/2. Similar (SSS), k = 1/2.
2
Are the triangles with sides 6, 8, 9 and 9, 12, 13 similar?
6 ÷ 9 = 8 ÷ 12 = 2/3 but 9 ÷ 13 is not 2/3. Not similar.
3
△ABC: AB = 6, AC = 10, ∠A = 55°. △DEF: DE = 9, DF = 15, ∠D = 55°. Show similarity, and find BC if EF = 12.
6 ÷ 9 = 10 ÷ 15 = 2/3 with equal included angles: SAS. BC = 12 × 2/3 = 8.
4
△ABC has AB = 6, BC = 9, ∠A = 50°; △DEF has DE = 4, EF = 6, ∠D = 50°. Can we say they are similar?
No. The angle ∠A is not between AB and BC, so this is SSA, which is not a similarity case.
5
Triangles with sides 8, 10, 12 and 4, 5, 6 are similar. Find k from the first to the second and the ratio of the perimeters.
