Determining Precisely Which Angles of a Triangle Are Congruent

[Bashyboy]

Hello,

Suppose that I have two triangles $\bigtriangleup ABC$ and $\bigtriangleup XYZ$ that are known to be congruent by the side-side-side axiom, from which it follows that the parts are also congruent, such as the angles. My question is, how do I determine which of the three angles of $\bigtriangleup ABC$ is congruent to ∠ A, for example? Visually, it is clear that $\angle A \cong X$, but I am having difficulty justifying this? Do I perform isometries until the vertices align, and then I can infer precisely which angles of $\bigtriangleup ABC$ are congruent to $\bigtriangleup ABC$?

 

[micromass]

Hello,

Suppose that I have two triangles $\bigtriangleup ABC$ and $\bigtriangleup XYZ$ that are known to be congruent by the side-side-side axiom,

I didn't know the side-side-side property was an axiom? It can definitely be proven! (Well, I suppose one could always make it an axiom, but I don't see the benefits in that).

Anyway, if side a and side b are congruent to side a' and side b', then the angle between ab is congruent to the angle between a'b'. This should answer your question.

 

[HallsofIvy]

Since you say you know that side BC, opposite to angle A is congruent to side YZ, opposite to angle X then it follows that angle A is congruent to angle X.