Triangle Larger Angle Theorem

The method of indirect proof will be used. First, the hypothesis and the conclusion of the theorem need to be identified. If m∠ A>m∠ C, then BC>AB. To start an indirect proof, the conclusion of the theorem needs to be assumed to be false. Assumption: BC≤ AB Next, any consequences of the assumption will be investigated. If a contradiction to the hypothesis is obtained, then the conclusion must be true. The assumption can be split into two parts. Assumption: BC=AB or BC

BC=AB

If BC equals AB, then the following can be concluded about △ ABC.

Claim	Explanation
BC=AB	Assumption
∠ A≅ ∠ C	Isosceles Triangle Theorem
m∠ A= m∠ C	Congruent angles have the same measure

This contradicts the hypothesis, which states that the measure of ∠ A is greater than the measure of ∠ C.

BC

If BC is less than AB, the following can be concluded about △ ABC.

Claim	Explanation
BC	Assumption
m∠ A< m∠ C	Triangle Longer Side Theorem

Again, this contradicts the hypothesis, which states that the measure of ∠ A is greater than the measure of ∠ C.

Conclusion

The assumption that BC is less than or equal to AB contradicts the hypothesis. Therefore, this assumption must be false. Consequently, the initial conclusion of the theorem is true.

BC>AB

It has been proven that if one angle of a triangle has a greater measure than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Proof

BC=AB

BC

Conclusion