Determining Precisely Which Angles of a Triangle Are Congruent

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Bashyboy
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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##?

congruent triangles.png
 

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Bashyboy said:
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.
 
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.