44
Relations between the sides and angles of a triangle
Textbook: pp. 112–113
GoalKnow the relation between the sides and angles of a triangle.
New words
larger side · katta tomonopposite angle · qarshisidagi burchakconverse theorem · teskari teoremaexterior angle · tashqi burchak
Explanation
Theorem: in a triangle the larger angle lies opposite the larger side. Proof: let AB > AC. On ray AB lay off AD = AC; D lies inside AB. Triangle △ACD is isosceles, so ∠ADC = ∠ACD. Also ∠ACB > ∠ACD since CD passes inside the angle. And ∠ADC is an exterior angle of △CDB, so ∠ADC > ∠B. Hence ∠ACB > ∠B. Converse: in a triangle the larger side lies opposite the larger angle. Corollary: equal angles lie opposite equal sides and vice versa. These facts are used to compare sides or angles.
Worked examples
In △ABC with AB = 7, BC = 5, AC = 9, the largest side is AC, so the largest angle is ∠B (opposite AC).
In △ABC with ∠A = 50°, ∠B = 60°, ∠C = 70°, the largest angle is ∠C, so the largest side is AB; the smallest side is BC (opposite ∠A).
Class activity
“Order the sides”: three stick lengths are given; pupils order the triangle’s angles from largest to smallest.
Practice
1
In △ABC, ∠A = 35°, ∠B = 95°, ∠C = 50°. Which side is longest?
AC (opposite ∠B)
2
In △ABC, AB = 8, BC = 6, AC = 10. Which angle is smallest?
∠A (opposite BC)
3
If two angles of a triangle are equal, what can be said about the opposite sides?
They are equal.
4
Why is the hypotenuse the longest side of a right triangle?
The right angle is the largest angle (the others are acute), and the largest side lies opposite it.