Third Angles Theorem

To begin, apply the Triangle Angle Sum Theorem to both triangles.

m ∠ A + m ∠ B + m ∠ C = 18 0^{\circ} m ∠ D + m ∠ E + m ∠ F = 18 0^{\circ}

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 = 18 0^{\circ} m ∠ A + m ∠ E + m ∠ C = 18 0^{\circ}

Next, subtract the second equation from first.

m ∠ A + m ∠ B + m ∠ C^{-} m ∠ A + m ∠ E + m ∠ C m ∠ B - m ∠ E = 18 0^{\circ} = 18 0^{\circ} = 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
$∠ A ≅ ∠ D$ and $∠ C ≅ ∠ F$	Given
$m ∠ A = m ∠ D$ and $m ∠ C = m ∠ F$	Definition of congruence
$m ∠ A + m ∠ B +$ $m ∠ C$ $= 18 0^{\circ}$ and $m ∠ D + m ∠ E +$ $m ∠ F$ $= 18 0^{\circ}$	Triangle Angle Sum Theorem
$m ∠ A + m ∠ B +$ $m ∠ C$ $= 18 0^{\circ}$ and $m ∠ A + m ∠ E +$ $m ∠ C$ $= 18 0^{\circ}$	Substitution
$m ∠ B - m ∠ E = 0$	Subtracting both equations
$m ∠ B = m ∠ E$	Addition Property of Equality
$∠ B ≅ ∠ E$	Definition of congruence

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