Question

Akira gave a problem to her sister Kiara. However, Kiara got stuck. Help Kiara identify whether the triangles are congruent and choose the correct option.
A. Yes, $\Delta ABC \cong \Delta DCE$
B. No, they are not congruent
C. Yes, $\Delta DCE \cong \Delta BAC$
D. Yes. $\Delta DCE \cong \Delta CAB$

Answer 1

Hint: According to the given figure, we will that see the sides are equal. Therefore we will take two triangles $\Delta CAB\,and\,\,\Delta DEC$ and will prove by SSS Criterion.

Complete step-by-step answer:

In the given figure
$AB = BE = EC = MD = MC$
$AC = DE$
And $DC = DM + MC$ …… (i)
Also $BC = BE + EC$ …… (ii)
Now, from equation (ii) we will replace $BE = MD$and $EC = MC$ then
$BC = MD + MC$ ….. (iii)
$DC = DM + MC$ ……. (iv)
Then from equation (iii) and (iv), we will get
$BC = DC$
Now, in $\Delta DCE\,\,and\,\,\Delta CAB,$
$DC = CB$ (proved above)
$DE = CA$ (given)
$CE = AB$ (given)
$\therefore \Delta DCE \cong \Delta CAB$ (SS Criterion)
Hence, the correct option is D.
Additional information: There are five ways to find if two triangles are congruent.
1. SSS: - We have two triangles with all three sides equal.
2. SAS: - We have two triangles where we know two sides and the included angle are equal.
3. ASA: - We have two triangles where we know two angles and the included sides are equal.
4. AAS: We have two triangles where we know two angles and the non-included sides are equal.
5. RHS: - We have two right-angled triangles with the same length of hypotenuse and same lengths for one of the other two sides.

Note: Students must keep in mind that if three sides of one triangle are equal to the corresponding three sides of another triangle then both triangles are congruent by SSS rule.