Equilateral and isosceles tricks

Equilateral and isosceles triangles are special because of their symmetry. That symmetry lets us deduce angles almost instantly.

Idea

Equilateral triangle (all three sides equal).

• All three angles are equal.
• By angle sum, each is $\tfrac{180^\circ}{3} = 60^\circ$.
• Every exterior angle is $120^\circ$.
• The altitude from any vertex bisects the opposite side and the angle, and divides the triangle into two congruent right triangles.

Isosceles triangle (two sides equal).

• The two angles opposite the equal sides are equal (called base angles).
• The third angle (between the equal sides) is the apex or vertex angle.
• Apex angle $+ 2 \times$ base angle $= 180^\circ$.
• The altitude from the apex bisects both the base and the apex angle.

Useful shortcuts.

• If apex $= 80^\circ$, each base angle $= (180 - 80)/2 = 50^\circ$.
• If a base angle $= 75^\circ$, the apex $= 180 - 2 \times 75 = 30^\circ$.

Combined classification reminders.

• A right isosceles triangle has angles $45^\circ, 45^\circ, 90^\circ$ (the two equal sides are the legs).
• A right scalene with $30^\circ$–$60^\circ$–$90^\circ$ angles is half of an equilateral triangle.

Worked examples

Example 1. In isosceles triangle $ABC$ with $AB = AC$, $\angle B = 65^\circ$. Find $\angle A$.

Base angles equal: $\angle C = 65^\circ$. Apex: $\angle A = 180 - 65 - 65 = 50^\circ$.

Example 2. In isosceles triangle, apex $= 100^\circ$. Find each base angle.

$(180 - 100)/2 = 40^\circ$ each.

Example 3. A triangle has all angles equal. Show each is $60^\circ$.

Let each be $x$. Then $3x = 180 \Rightarrow x = 60^\circ$. So angles are all $60^\circ$. (Such a triangle is also equilateral by the converse.)

Example 4. An isosceles triangle has a right angle. Find the other two angles.

The right angle is between the two equal sides (apex). The other two are base angles, summing to $90^\circ$ and being equal , so $45^\circ$ each.

Try it yourself

1. Equilateral triangle: each angle is?
2. Isosceles with apex $40^\circ$: base angles?
3. Isosceles with base angle $50^\circ$: apex?
4. Right isosceles with right angle as apex: other angles?
5. Isosceles triangle with apex $90^\circ$ , give an example real-life shape.
6. If an angle of an isosceles triangle is $30^\circ$, what is the largest its apex can be?
7. Sides of equilateral triangle are doubled. Are the angles also doubled?
8. The three exterior angles of an equilateral triangle each measure?

Activity

Fold a square piece of paper along its diagonal , you get a right isosceles triangle. Measure all three angles with a protractor.

Cut out three identical equilateral triangles and try to make them tile a flat region. (You will see they fit perfectly around a point, which is why they work in tilings.)