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Rule

Third Angles Theorem

If two angles of a given triangle are congruent with two angles of another triangle, then the third angles of each triangle are also congruent.
Two triangles ABC and DEF with two pairs of congruent angles

Based on the diagram, the following relation holds true.

If and then

Proof

To begin, apply the Triangle Angle Sum Theorem to both triangles.
Now, since it is given that and substitute for and for into the second equation.
Next, subtract the second equation from first.
Finally, solve the equation above for

With this, by the definition of congruence. The above proof is organized in a two-column proof table below.

Statements Reasons
and Given
and Definition of congruence
and Triangle Angle Sum Theorem
and Substitution
Subtracting both equations
Addition Property of Equality
Definition of congruence