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Rule

# Third Angle 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.

Based on the diagram, the following relation holds true.

If $\angle A \cong \angle D$ and $\angle C \cong \angle F,$ then $\angle B \cong \angle E.$

### Proof

To begin, apply the Triangle Angle Sum Theorem to both triangles. $\begin{gathered} m\angle A + m\angle B + m\angle C = 180 ^\circ \\ m\angle D + m\angle E + m\angle F = 180 ^\circ \end{gathered}$ Now, since it is given that $\angle A \cong \angle D$ and $\angle C \cong \angle F,$ substitute $m\angle {\color{#0000FF}{A}}$ for $m\angle D,$ and $m\angle {\color{#009600}{C}}$ for $m\angle F.$ $\begin{gathered} m\angle A + m\angle B + m\angle C = 180 ^\circ \\ m\angle {\color{#0000FF}{A}} + m\angle E + m\angle {\color{#009600}{C}} = 180 ^\circ \end{gathered}$ Subtract the second equation from first one. \begin{aligned} \cancel{m\angle A} + m\angle B + \cancel{m\angle C} &= 180 ^\circ \\ ^{\large{-}\,\,} \cancel{m\angle A} + m\angle E + \cancel{m\angle C} &= 180 ^\circ \\\hline m\angle B - m\angle E &= 0 \end{aligned} Finally, solve the equation above for $m\angle B.$

$\begin{gathered} m\angle B - m\angle E = 0 \\ \Downarrow \\ m\angle B = m\angle E \end{gathered}$

With this, $\angle B \cong \angle E$ by definition of congruence. The above proof is organized in a two-column proof table below.

 Statements Reasons $\angle A \cong \angle D$ and $\angle C \cong \angle F$ Given $m\angle A = m\angle D$ and $m\angle C = m\angle F$ Definition of congruence $m\angle A + m\angle B \,+$ $m\angle C$ $=180 ^\circ$ and $m\angle D + m\angle E \,+$ $m\angle F$ $=180^\circ$ Triangle Angle Sum Theorem $m\angle A + m\angle B \,+$ $m\angle C$ $=180 ^\circ$ and $m\angle D + m\angle E \,+$ $m\angle C$ $=180^\circ$ Substitution $m\angle B - m\angle E = 0$ Subtracting both equations $m\angle B = m\angle E$ Addition Property of Equality $\angle B \cong \angle E$ Definition of congruence