Two angles are enough for triangles: the AA postulate and the special cases of equilateral, isosceles and right triangles
Mathematics Sec 1 Term 1, Unit 2, Lessons 1–2
GoalProve that two triangles are similar from two equal angles and use the similarity to find lengths.
New words
AA postulate · if two angles of one triangle equal two angles of another, the triangles are similarshadow method · using the sun’s parallel rays to make similar triangles and find a height
Explanation
For triangles we do not need to check everything. If two angles of one triangle are equal to two angles of another, the third angles are equal too (the angles of a triangle add to 180°), and the triangles are similar. This is the AA postulate. Special cases: all equilateral triangles are similar; two isosceles triangles are similar if their vertex angles are equal or if a base angle of one equals a base angle of the other; two right triangles are similar if an acute angle of one equals an acute angle of the other. Then use the order of the letters to write the proportion. Careful: knowing only that sides are proportional is another test (next week), and one equal angle alone is not enough in general. Also, if a line DE is parallel to BC then ∠ADE = ∠ABC (corresponding angles) and ∠A is common, so △ADE ~ △ABC.
Worked examples
△ABC has ∠A = 50° and ∠B = 60°. △DEF has ∠D = 50° and ∠F = 70°. Then ∠C = 180° − 110° = 70° and ∠E = 180° − 120° = 60°. All three angles match, so △ABC ~ △DEF with A→D, B→E, C→F.
In △ABC, D is on AB and E is on AC with DE ∥ BC, AD = 6, DB = 3 and DE = 8. Find BC and AC if AE = 4. ∠A is common and ∠ADE = ∠ABC, so △ADE ~ △ABC. AB = 9, so AD/AB = 6/9 = 2/3 = DE/BC = AE/AC. So BC = 8 ÷ 2/3 = 12 and AC = 4 ÷ 2/3 = 6.
A 1.5 m stick casts a 2 m shadow. At the same time a palm tree casts a 14 m shadow. The sun’s rays are parallel, so the two right triangles share an acute angle and are similar: height/14 = 1.5/2, so the height is 14 × 1.5/2 = 10.5 m.
Class activity
Find the height. In the sun, measure the length of your own shadow and of a lamp post or a tree’s shadow at the same moment. Write the proportion and find the height of the post, if you know your own height.
Practice
1
Triangles ABC and DEF: ∠A = 45°, ∠B = 80°, and ∠D = 80°, ∠F = 55°. Are they similar? Write the similarity.
∠C = 55° and ∠E = 45°. So A→E, B→D, C→F: △ABC ~ △EDF (not △DEF!).
2
Two right triangles each have an angle of 35°. Are they similar? Why?
Yes: each has a right angle and a 35° angle, so two angles are equal (AA). The third angles are both 55°.
3
In △ABC, DE ∥ BC with D on AB, E on AC, AD = 4, DB = 2 and DE = 6. Find BC.
△ADE ~ △ABC, AB = 6, so 4/6 = 6/BC and BC = 9.
4
A 2 m post casts a 3 m shadow. At the same moment a building casts a 24 m shadow. How tall is the building?
h/24 = 2/3, so h = 16 m.
5
A student says: “Two isosceles triangles are always similar.” Is that true? Give an example.
No. An isosceles triangle with vertex angle 40° and one with vertex angle 100° have different angles, so they are not similar.
