Question

In the figure below, ZABC = ZDEC and ZGFE - ZDCE. Point C is the point of intersectionbetween AG and BD, while point E is the point of intersection between AG and DFProve ABC - GEF

Answer 1

Given that

$\begin{gathered} \angle ABC\cong\angle DEC \\ \angle GFE\cong\angle DEC \end{gathered}$

C is the point of intersection between AG and BD while E is the point of intersection between AG and DF

To prove the triangle ABC is similar to the triangle GEF

From the given figure, it is seen that

$CD=CE$

Therefore,

$\angle DEC=\angle DCE$

Since

$\begin{gathered} \angle ABC\cong\angle DEC \\ \angle GFE\cong\angle DEC \end{gathered}$

Then,

$\angle ABC\cong\angle GFE$

In the triangle ABC and the triangle GFE, two angles are congruent.

Therefore,

$\begin{gathered} \frac{AC}{BC}=\frac{EG}{FE} \\ \frac{AC}{EG}=\frac{BC}{FE} \end{gathered}$

Similarly,

$\frac{AB}{FG}=\frac{AC}{EG}$

Thus,

$\frac{AB}{FG}=\frac{AC}{EG}=\frac{BC}{FE}$

Hence, the triangle ABC is similar to the triangle GEF