Like the judicial system, geometry is a field that requires statements to be proved before they can be applied. In a court of law, we use evidence to prove or disprove the plaintiff's conjecture. In mathematics, we use theorems and properties that have already been proved to help us arrive at a logical conclusion about a conjecture.
For example, if you know that two triangles are congruent, you can assume that the following two statements are true about the triangles.
The triangles have three sets of congruent (of equal length) sides.
The triangles have three sets of congruent (of equal measure) angles.
Proving that the two triangles are, in fact, congruent does not require us to show that both statements are true. If we know that all three sides of two different triangles have the same length, then we can know that the triangles are congruent using the SSS (side-side-side) theorem—without even knowing any of the angle measurements.
Suppose we apply this type of reasoning to the ice cream example. If your parents tested the DNA on the spoon left in the empty ice cream bucket and it matched your DNA, they could prove you ate the ice cream without even knowing you stayed up late to watch a movie.
Question
What can you say about angles of two triangles once you have determined that their sides have the same length?
