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Third Angle Theorem


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 AD\angle A \cong \angle D and CF,\angle C \cong \angle F, then BE.\angle B \cong \angle E.


To begin, apply the Triangle Angle Sum Theorem to both triangles. mA+mB+mC=180mD+mE+mF=180\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 AD\angle A \cong \angle D and CF,\angle C \cong \angle F, substitute mAm\angle {\color{#0000FF}{A}} for mD,m\angle D, and mCm\angle {\color{#009600}{C}} for mF.m\angle F. mA+mB+mC=180mA+mE+mC=180\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. mA+mB+mC=180mA+mE+mC=180mBmE=0\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 mB.m\angle B.

mBmE=0mB=mE\begin{gathered} m\angle B - m\angle E = 0 \\ \Downarrow \\ m\angle B = m\angle E \end{gathered}

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

Statements Reasons
AD\angle A \cong \angle D and CF\angle C \cong \angle F Given
mA=mDm\angle A = m\angle D and mC=mFm\angle C = m\angle F Definition of congruence
mA+mB+m\angle A + m\angle B \,+ mCm\angle C =180=180 ^\circ and mD+mE+m\angle D + m\angle E \,+ mF m\angle F =180=180^\circ Triangle Angle Sum Theorem
mA+mB+m\angle A + m\angle B \,+ mCm\angle C =180=180 ^\circ and mD+mE+m\angle D + m\angle E \,+ mCm\angle C =180=180^\circ Substitution
mBmE=0m\angle B - m\angle E = 0 Subtracting both equations
mB=mEm\angle B = m\angle E Addition Property of Equality
BE\angle B \cong \angle E Definition of congruence