☰ Year plan · Mathematics

Triangles that cut off similar triangles, and the SSS and SAS tests

Week 6 · Similar triangles: the parallel-line corollary, the altitude to the hypotenuse, and the SSS and SAS cases · 3 lessons
3

Proportional sides (SSS) and one equal angle between proportional sides (SAS): proving that two triangles are similar

Mathematics Sec 1 Term 1, Unit 2, Lesson 2
GoalProve that two triangles are similar from their sides (SSS) or from two sides and the included angle (SAS).
New words
SSS · all three pairs of corresponding sides are proportionalSAS · two pairs of sides are proportional and the angles between them are equalincluded angle · the angle formed by the two sides you are using
Explanation

There are two more tests, besides AA. SSS: if AB/DE = BC/EF = CA/FD, then △ABC ~ △DEF, and the equal angles follow. Compare the shortest with the shortest, the longest with the longest. SAS: if one angle of a triangle equals one angle of another and the sides that include these angles are proportional, the triangles are similar. The angle must lie BETWEEN the two sides; two proportional sides and an equal angle that is not between them prove nothing. A common use of SAS: if AC² = CD × CB then AC/CB = CD/AC; together with the common angle C this gives △ACD ~ △BCA. Remember to name the vertices in matching order.

TestWhat you check
SSSall three pairs of sides have the same ratio
SAStwo pairs of sides have the same ratio and the angles BETWEEN them are equal
AAtwo pairs of angles are equal
Worked examples
△ABC has AB = 6, BC = 9, CA = 12 and △DEF has DE = 8, EF = 12, FD = 16. 6/8 = 9/12 = 12/16 = 3/4, so △ABC ~ △DEF by SSS with k = 3/4. Not similar: sides 4, 6, 9 and 6, 9, 12 give 4/6 = 6/9 = 2/3 but 9/12 = 3/4.
D is on AB and E is on AC of △ABC with AB = 12, AC = 18, AD = 8 and AE = 12. AD/AB = 8/12 = 2/3 and AE/AC = 12/18 = 2/3, and ∠A is common (between the sides), so △ADE ~ △ABC by SAS. So ∠ADE = ∠ABC, which are corresponding angles, and therefore DE ∥ BC. Also DE/BC = 2/3.
In △ABC, D is on BC with AC = 6, CD = 4 and CB = 9. Then AC² = 36 = CD × CB. So AC/CB = 6/9 = 2/3 = CD/AC = 4/6, and ∠C is common, so △ACD ~ △BCA by SAS. It follows that ∠CAD = ∠CBA.
Class activity

Two tests, one triangle. Draw △ABC with AB = 6, BC = 9 and CA = 12 cm, and measure its angles. Then draw △DEF with all sides multiplied by 4/3 and compare the angles.

Practice
1
Are triangles with sides 5, 12, 13 and 10, 24, 26 similar?
2
Are triangles with sides 3, 4, 6 and 6, 8, 11 similar?
3
△ABC: AB = 9, AC = 6, ∠A = 40°. △DEF: DE = 6, DF = 4, ∠D = 40°. Similar?
4
Two triangles have two pairs of proportional sides and one pair of equal angles, but the angle lies outside those sides. Can we say they are similar?
5
D and E are the midpoints of AB and AC in △ABC and BC = 14 cm. Find DE.