Third Angles Theorem - Rule, Proof and Examples

To begin, apply the Triangle Angle Sum Theorem to both triangles. m∠ A + m∠ B + m∠ C = 180 ^(∘) m∠ D + m∠ E + m∠ F = 180 ^(∘) Now, since it is given that ∠ A ≅ ∠ D and ∠ C ≅ ∠ F, substitute m∠ A for m∠ D and m∠ C for m∠ F into the second equation. m∠ A + m∠ B + m∠ C = 180 ^(∘) m∠ A + m∠ E + m∠ C = 180 ^(∘) Next, subtract the second equation from first. m∠ A + m∠ B + m∠ C &= 180 ^(∘) ^- m∠ A + m∠ E + m∠ C &= 180 ^(∘) m∠ B - m∠ E &= 0 Finally, solve the equation above for m∠ B.

m∠ B - m∠ E = 0 ⇓ m∠ B = m∠ E

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

Statements	Reasons
1. ∠ A ≅ ∠ D and ∠ C ≅ ∠ F	1. Given
2. m∠ A = m∠ D and m∠ C = m∠ F	2. Definition of congruence
3. m∠ A + m∠ B + m∠ C =180 ^(∘) and m∠ D + m∠ E + m∠ F =180^(∘)	3. Triangle Angle Sum Theorem
4. m∠ A + m∠ B + m∠ C =180 ^(∘) and m∠ A + m∠ E + m∠ C =180^(∘)	4. Substitution
5. m∠ B - m∠ E = 0	5. Subtracting both equations
6. m∠ B = m∠ E	6. Addition Property of Equality
7. ∠ B ≅ ∠ E	7. Definition of congruence