Isosceles and equilateral angles

Congruence is a powerful tool for discovering properties of figures, not just for comparing two drawings. Here it is at work.

Take an isosceles triangle ∆ABC with AB = AC. What can we say about ∠B and ∠C? Construct the ALTITUDE from A down to BC, meeting it at D.

Now list what is known about ∆ADB and ∆ADC. AB = AC, given. ∠ADB = ∠ADC = 90°, from the construction. And AD is a COMMON side of both triangles.

That is a right angle, the hypotenuse and one other side — the RHS condition. So ∆ADB ≅ ∆ADC.

Congruent triangles have equal corresponding parts, and ∠B and ∠C correspond. Therefore ∠B = ∠C. Stated generally: The angles opposite to equal sides are equal. For example, if ∠A were 40°, the other two would share the remaining 140°, so each would be 70°.

Now an equilateral triangle, where all three sides are equal. Apply the new fact twice. Since AB = AC, the angles opposite them are equal: ∠B = ∠C. Since AB = BC, likewise ∠A = ∠C.

So all three angles of an equilateral triangle are equal, just as its sides are. And what are they? The three angles add to 180°, so 3 × (one angle) = 180°, and each angle is 60°. Using nothing but the notion of congruence, we have deduced that every equilateral triangle has three 60° angles. Congruent triangles turn up everywhere in design and construction, from the Pyramid of Giza to the Louvre in Paris to a rangoli pattern on a doorstep.