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Based on the diagram above, the following relation holds true.
m∠ A>m∠ C ⇒ BC>AB
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
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.
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.
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.