Proof of the converse of Theorem 7
We use proof by contradiction (students at higher level leaving cert are expected to know the meaning of this term)
(Note on proof by contradiction: Assume a statement is not true and show that this assumption leads to a contradiction – called reduction as absurdum (reduction to absurdity) in Latin.)
To Prove: The side opposite the greater of two angles in a triangle is greater than the side opposite the lesser of two angles.
Given: |<ABC| > |<ACB|
To Prove: |AC| > |AB|
Proof: Assuming that |AC| is not greater than |AB|, what are the only other options for the relationship between |AC| and |AB|?
Option 1: _________________________
Option 2: _________________________
If option 1 is true draw the triangle which would represent option 1.
Hence what type of triangle is triangle ABC? _____________
Hence what is the relationship between the |<ABC| and |<ACB|? _________________
Is this in agreement with or does it contradict, what we were given? ___________________
Hence, can option 1, i.e. __________________, be true? _______
If option 2 is true draw the triangle which would represent option 2.
Using the theorem we proved earlier, what does this does this tell us about the relationship between
|<ABC| and |<ACB| in this scenario? ________________________
Is this in agreement with or does it contradict, what we were given? _______________________
Hence, can option 2, i.e. ____________________________ , be true? ______
If there are only 3 options, which option/s are now possible for the relationship between |AC| and
|AB| given that |<ABC| > |<ACB|? ____________________________
* Taken from projectmaths document: www.projectmaths.ie/documents/teachers/theorem_7.pdf