Sides and angles of a triangle

In a triangle, equal sides face equal angles, and equal angles face equal sides.

Today: when the sides are different, the larger side faces the larger angle.

Theorem: if two sides of a triangle are different, the larger of them faces the larger angle.

• In triangle $\displaystyle ABC$, if $\displaystyle AC>AB$, then $\displaystyle \angle B>\angle C$.
• $\displaystyle AC$ faces $\displaystyle \angle B$, and $\displaystyle AB$ faces $\displaystyle \angle C$.

Given: $\displaystyle AC>AB$.
Prove: $\displaystyle \angle ABC>\angle ACB$.

The proof below marks the point $\displaystyle D$ on $\displaystyle AC$ so that $\displaystyle {AD}={AB}$: it is inside the side, since $\displaystyle AC>AB$. Triangle $\displaystyle ABD$ is isosceles, so $\displaystyle {\angle ABD}={\angle ADB}$, and their size lies between the two angles being compared:

• angle $\displaystyle ABD$ is a part of angle $\displaystyle ABC$, so it is smaller;
• angle $\displaystyle ADB$ is an exterior angle of triangle $\displaystyle BDC$, so it is larger than angle $\displaystyle C$.

Theorem: if two angles of a triangle are different, the larger of them faces the larger side.

In triangle $\displaystyle ABC$, $\displaystyle \angle B>\angle C$.
The proof below compares $\displaystyle AC$ with $\displaystyle AB$: of the three possibilities, it rules out two.

• if $\displaystyle {AC}={AB}$, the triangle would be isosceles, and its base angles would be equal – against what is given;
• if $\displaystyle AC<AB$, by the theorem angle $\displaystyle B$ would be smaller than angle $\displaystyle C$ – against what is given;
• only $\displaystyle AC>AB$ is left.

The order of a triangle’s angles is the order of the sides facing them: the largest angle faces the largest side, and the smallest the smallest.

In triangle $\displaystyle ABC$, $\displaystyle {\angle A}={70^{\circ}}$ and $\displaystyle {\angle B}={45^{\circ}}$.

• $\displaystyle {\angle C}={180^{\circ}-70^{\circ}-45^{\circ}}={65^{\circ}}$, so $\displaystyle \angle A>\angle C>\angle B$;
• the sides facing them, in the same order: $\displaystyle BC>AB>AC$.

The sides of triangle $\displaystyle KLM$ are $\displaystyle {KL}={4,\ LM}={6,\ KM}={7}$.

• The sides from the largest to the smallest: $\displaystyle KM>LM>KL$;
• the angles facing them, in the same order: $\displaystyle \angle L>\angle K>\angle M$.

Order the sides of the triangle from the largest to the smallest, and give the reasons.

In triangle $\displaystyle ABC$, $\displaystyle {\angle A}={50^{\circ}}$ and $\displaystyle {\angle B}={60^{\circ}}$.

In triangle $\displaystyle PQR$, $\displaystyle {\angle P}={100^{\circ}}$, and angle $\displaystyle Q$ is 3 times angle $\displaystyle R$.

Order the angles of the triangle from the largest to the smallest, and give the reasons.

The sides of triangle $\displaystyle DEF$ are $\displaystyle {DE}={5,\ EF}={9,\ DF}={7}$.

In triangle $\displaystyle XYZ$, $\displaystyle {XY}={YZ}$ and $\displaystyle {\angle Y}={50^{\circ}}$.
Which side is the smallest?

In triangle $\displaystyle ABC$, angle $\displaystyle C$ is right.
Prove: $\displaystyle AB>AC$ – the hypotenuse is longer than a leg.

In triangle $\displaystyle ABC$, the segment $\displaystyle AD$ bisects angle $\displaystyle A$.
Prove: $\displaystyle AB>BD$.