Isosceles triangles and inequalities

The isosceles triangle

A triangle with two equal sides is called isosceles. The two equal sides are its legs, the third side is its base, and the angle between the legs is its apex angle. The two angles at the ends of the base are the base angles.

Theorem 1. Angles opposite equal sides of a triangle are equal. In an isosceles triangle, the two base angles are equal.

Proof. In triangle $ABC$, let $AB = AC$. Draw the bisector $AD$ of the angle $A$, meeting $BC$ at $D$.

1. $AB = AC$. This is given.
2. $\angle BAD = \angle CAD$. The line $AD$ bisects $\angle A$.
3. $AD = AD$. This is the common side.
4. So $\triangle ABD \cong \triangle ACD$, by SAS.
5. Therefore $\angle B = \angle C$, by CPCT.

Theorem 2 (the converse). Sides opposite equal angles of a triangle are equal.

Proof. In triangle $ABC$, let $\angle B = \angle C$. Draw the perpendicular $AD$ from $A$ to $BC$.

1. $\angle B = \angle C$. This is given.
2. $\angle ADB = \angle ADC = 90^\circ$. The line $AD$ is perpendicular to $BC$.
3. $AD = AD$. This is the common side.
4. So $\triangle ADB \cong \triangle ADC$, by AAS.
5. Therefore $AB = AC$, by CPCT.

The perpendicular from the apex

In the isosceles triangle $ABC$ with $AB = AC$, let $AD$ be the perpendicular from $A$ to the base $BC$.

The triangles $ABD$ and $ACD$ have a right angle at $D$, equal hypotenuses $AB = AC$ and a common side $AD$. They are congruent by RHS. By CPCT, $BD = DC$ and $\angle BAD = \angle CAD$. So the perpendicular from the apex bisects the base and the apex angle.

An equilateral triangle has three equal sides. Using Theorem 1 on each pair of sides, all three angles are equal, so each one is $60^\circ$. Conversely, a triangle with three equal angles has three equal sides.

A worked example. In triangle $ABC$ with $AB = AC$, the angle $B$ is $(6y - 7)^\circ$ and the angle $C$ is $(4y + 13)^\circ$. Find $y$ and the angle $A$.

1. Equal base angles. $\angle B = \angle C$, so $6y - 7 = 4y + 13$.
2. Solve. $2y = 20$, so $y = 10$.
3. The base angles. $\angle B = \angle C = 53^\circ$.
4. The apex. $\angle A = 180^\circ - 2 \times 53^\circ = 74^\circ$.

Unequal sides, unequal angles

Theorem 3. If two sides of a triangle are unequal, the angle opposite the longer side is larger.

Proof. In triangle $ABC$, let $AC > AB$. Mark the point $D$ on $AC$ with $AD = AB$.

1. In triangle $ABD$, $AB = AD$, so $\angle ABD = \angle ADB$ by Theorem 1.
2. The angle $ADB$ is an exterior angle of triangle $BDC$, so it is larger than $\angle C$.
3. Also $\angle ABC$ is larger than $\angle ABD$.
4. So $\angle ABC > \angle ABD = \angle ADB > \angle C$. The larger angle $B$ lies opposite the longer side $AC$.

Theorem 4 (the converse). In a triangle, the side opposite the larger angle is longer.

If $AC$ were not longer than $AB$, it would be equal to $AB$, and then the angles $B$ and $C$ would be equal. Or $AC$ would be shorter, and then $\angle B$ would be smaller than $\angle C$. Both are against the given. So the side opposite the larger angle is the longer one.

The triangle inequality

Theorem 5. The sum of any two sides of a triangle is greater than the third side.

Proof. In triangle $ABC$, extend $BA$ to a point $D$ with $AD = AC$. Join $C$ to $D$.

1. In triangle $ACD$, $AD = AC$, so $\angle ACD = \angle ADC$.
2. The angle $BCD$ is larger than $\angle ACD$, so $\angle BCD > \angle BDC$.
3. In triangle $BCD$, the side opposite the larger angle is longer: $BD > BC$.
4. But $BD = BA + AD = BA + AC$. So $AB + AC > BC$.

This is why a straight path is the shortest. Walking from $B$ to $C$ along the sides $BA$ and $AC$ is always longer than walking straight.

Which sides make a triangle? Take the longest side. It must be less than the sum of the other two. Can $3$, $4$ and $8$ be the sides of a triangle? Here $3 + 4 = 7$, which is less than $8$, so no. What about $5$, $6$ and $10$? Here $5 + 6 = 11$ is more than $10$, so yes.

The third side. If two sides are $a$ and $b$, the third side $c$ must satisfy $|a - b| < c < a + b$. With $a = 8$ and $b = 3$, the third side lies between $5$ and $11$ and cannot be $5$ or $11$.

Practise with new numbers

Try it in Python

This program checks five sets of three sticks and says which ones can form a triangle.

Common mistakes

• Using the apex angle as a base angle. In an isosceles triangle with $AB = AC$, the equal angles are $B$ and $C$, not $A$.
• Allowing a flat triangle. If the longest side equals the sum of the other two, the three sides lie on one line.
• Putting the larger angle opposite the shorter side. The longer side faces the larger angle.
• Checking only one pair of sides. The longest side is the one to test against the other two.

Remember these

• Equal sides have equal angles opposite them, and equal angles have equal sides opposite them.
• The perpendicular from the apex of an isosceles triangle bisects the base and the apex angle.
• The longer side is opposite the larger angle, and the larger angle is opposite the longer side.
• Any two sides together are longer than the third.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In triangle $PQR$, $PQ = PR$ and $\angle Q = 72^\circ$. Find $\angle P$.

2. A triangle has angles $50^\circ$, $50^\circ$ and $80^\circ$. Which sides are equal?

3. Can $4\ \text{cm}$, $9\ \text{cm}$ and $15\ \text{cm}$ be the sides of a triangle?

4. In triangle $ABC$, $\angle A = 40^\circ$ and $\angle B = 75^\circ$. Which side is the shortest?

5. Why can an equilateral triangle not have an angle of $70^\circ$?