Exercise 43 Page 342

Consider the given diagram.

Using this diagram, we will prove the Triangle Longer Side Theorem.

Triangle Longer Side Theorem

If one side of a triangle is longer than another side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.

Let's state exactly what is given and what we want to prove. Since the length of $\overline{BD}$ is equal to the length of $\overline{BA},$ by the definition of congruent segments, $\overline{BD}$ is congruent to $\overline{BA}.$ Then, using the Base Angles Theorem, we can conclude that $\angle 1$ is congruent to $\angle 2.$ By the definition of congruent angles, we can say that $m\angle 1=m\angle 2.$ Next, by the Angle Addition Postulate, we have $m\angle BAC=m\angle 1+m\angle 3.$ Then, using the Substitution Property of Equality, we can substitute $m\angle 1=m\angle 2$ into the equation for $m\angle BAC.$ From the diagram, we can see that $\angle 2$ is an exterior angle of $△ADC.$ Therefore, by the Triangle Exterior Angle Theorem, we know that $m\angle 2$ is the sum of $m\angle 3$ and $m\angle C.$ Let's now use the Transitive Property of Equality. Finally, we will solve this equation for $m\angle BAC.$ As we can see, $m\angle BAC$ is the sum of $2m\angle 3$ and $m\angle C.$ Note that all this measures are positive. Therefore, the obtained equation implies that $m\angle BAC>m\angle C.$