I) Theorems:
a) Hinge Theorem - if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.
Example: Given ΔRST and ΔVWX, RS = VW and ST = WX, angle S = 100 degrees and angle W = 80 degrees, we can conclude that RT is greater than VX.
B) Converse of the Hinge theorem - if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.
II) Using Indirect Proofs:
A) Indirect Proof - is a proof in which you prove that a statement is true by first assuming that its opposite is true. If this assumption leasds to an impossibility, then you have proved that the original statement is true.
1) Guidelines for writing an indirect proof:
a) identify the statement that you want to prove is true.
b) begin by assuming the statement is false; assum its opposite is true.
c) obtain statements that logically follow from your assumption.
d) if you obtain a contradiction, then the original statement must be true.
Example: Prove a triangle cannot have 2 right angles.
1) Given ΔABC.
2) Assume angle A and angle B are both right angles is true by one of two possibilities (it is either true or false so we assume it is true).
3) measure of angle A = 90 degrees and measure of angle B = 90 degrees by definition of right angles.
4) measure of angle A + measure of angle B + measure of angle C = 180 degrees by the sum of the angles of a triangle is 180 degrees.
5) 90 + 90 + measure of angle C = 180 by substitution.
6) measure of angle C = 0 degrees by subtraction postulate
7) angle A and angle B are both right angles is false by contradiction (an angle of a triangle cannot equal zero degrees)
8) A triangle cannot have 2 right angles by elimination (we showed since that if they were both right angles, the third angle would be zero degrees and this is a contridiction so therefore our assumption was false ).
Here is another direct proof in t-table form:
Example 2:
This is a direct proof.
Given: LM = MN
Prove: line segment LM is congruent to line segment MN.
This is example 2 as an indirect proof:
Example 3:
Given: LM = MNProve: line segment LM is congruent to line segment MN.
Example 4: Given angle PQR is a straight angle, prove the measure of the angle is equal to 180 degrees indirectly.
Example 5: Prove the following indirectly.
As you may have noticed, each indirect proof has 4 steps that have the same concepts:
1. given
2. Assume the opposite of what you want to prove is true.
2nd to last step, your assumption was false by contradiction.
last step, what you wanted to prove is proven by elimination.
