Theorem 1

Prove that: If two sides of a triangle are unequal in length, the longer side has an angle of greater measure opposite to it.
Given: In $\triangle ABC$, $m \overline{BC}$ >$m\overline{AC}$
To prove: $m\angle A \gt m\angle B$

Construction: From $\overline{BC}$ cutoff $$\overline{CD} \cong \overline{AC}$$. Join A and D

Explanation

To prove that the longer side of a triangle has an angle of greater measure opposite to it, we start by assuming we have a triangle ABC with two sides of unequal length. We want to show that angle A is greater than angle B.

To do this, we draw a line from point A to a point D on side AB, such that CD is congruent to AC. This creates a new triangle ACD that is congruent to triangle CBD.

Next, we use the fact that the exterior angle of a triangle is greater than any non-adjacent interior angle to show that angle CDA is greater than angle B.

Then, we use the fact that the angles of a triangle add up to 180 degrees to show that angle A is greater than angle CDA.

Finally, we use the transitive property of inequality to conclude that angle A is greater than angle B.

Example

Prove that: In a scalene triangle, the angle opposite to the largest side is of measure greater than $60\degree$.

Given: In $\triangle ABC$, with $m\overline{AC}$>$m\overline{AB}$ and $m\overline{AC}$>$m\overline{BC}$

To Prove: m\angle B > 60\degree

Explanation

• To prove that in a scalene triangle, the angle opposite to the largest side is of measure greater than $60\degree$, we start by examining a triangle $\triangle ABC$ with $m\overline{AC}$ > $m\overline{AB}$ and $m\overline{AC}$ > $m\overline{BC}$. We want to prove that $m\angle B$ > $60\degree$.

• To begin the proof, we note that in $\triangle ABC$, we have $m\angle B$ > $m\angle C$ and $m\angle B$ > $m\angle A$. This is because the side opposite to $m\angle B$ is the largest side in the triangle, as given in the problem.

• Next, we recall that the sum of the three angles in a triangle is $180\degree$. Therefore, we can write $m\angle A$ + $m\angle B$ + $m\angle C$ = $180\degree$.

• Using the fact that $m\angle B$ > $m\angle C$ and $m\angle B$ > $m\angle A$, we can substitute $m\angle B$ for both $m\angle C$ and $m\angle A$ in the equation above. This yields $3m\angle B$ > $180\degree$.

• By dividing both sides of the inequality by $3$, we obtain $m\angle B$ > $\frac{180\degree}{3}$, which simplifies to $m\angle B$ > $60\degree$.

• Therefore, we have shown that in a scalene triangle, the angle opposite to the largest side is of measure greater than $60\degree$.