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I have the following picture:

$AG = BF$

I only found $4$ pairs of congruent triangles: {$\triangle ACO$, $\triangle ODB$}; {$\triangle CFE$, $\triangle EDG$}; {$\triangle ODF$, $\triangle COG$}; {$\triangle AOC$, $\triangle CFE$}.

However, the answer said there should be $5$.

Where's the fifth pair of congruent triangles?

2 Answers 2

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This answer shows that if we are not allowed to draw extra lines, then there are only four pairs of congruent triangles.

(If we are allowed to draw extra lines, the number of such pairs is more than four since we have $\triangle{OAF}\equiv\triangle{OBG}$.)

$\qquad\qquad\qquad$enter image description here

Since we are not allowed to draw extra lines, we have only eight triangles to consider :

$$\triangle{ACO},\quad \triangle{AGO},\quad\triangle{DEG},\quad\triangle{BDO}$$ $$\triangle{BFO},\quad\triangle{DFO},\quad\triangle{CEF},\quad\triangle{CGO}$$

Considering only angles, there are only eight candidates for congruent pairs :

$$(\triangle{ACO},\triangle{GDE}),\quad (\triangle{ACO},\triangle{BDO}),\quad (\triangle{ACO},\triangle{FCE}),\quad (\triangle{AGO},\triangle{BFO})$$ $$(\triangle{GDE},\triangle{BDO}),\quad (\triangle{GDE},\triangle{FCE}),\quad (\triangle{BDO},\triangle{FCE}),\quad (\triangle{DFO},\triangle{CGO})$$

Now, we have $$\triangle{ACO}\equiv \triangle{BDO},\quad \triangle{GDE}\equiv\triangle{FCE},\quad \triangle{DFO}\equiv\triangle{CGO},\quad\triangle{AGO}\equiv\triangle{BFO}$$

However, we have $$\triangle{ACO}\not\equiv\triangle{GDE},\quad \triangle{ACO}\not\equiv\triangle{FCE},\quad \triangle{GDE}\not\equiv\triangle{BDO},\quad \triangle{BDO}\not\equiv\triangle{FCE}$$ since $$AO\not=GE,\quad AC\not=FC,\quad DG\not=DB,\quad BO\not=FE$$ respectively.

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    Instead of $OAF≡OBG$, how about $OCG≡ODF$? – 2017-02-26
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    @TakahiroWaki: The OP has already noticed $\triangle{ODF}\equiv \triangle{OCG}$. (see the first sentence in my answer.) – 2017-02-26
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    @mathlove you are not allowed to draw extra lines. – 2017-03-01
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    @Kw08: Then, there are only four pairs of congruent triangles. See the answer. – 2017-03-02
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The pair AOG and FOB are built out of triangles you have as congruent, so they are congruent.