Question

Which of the following pairs of triangles are congruent ? In each case, state the condition of congruency:

(a) In $\Delta ABC$ and $\Delta DEF$, $AB=DE,BC=EF$ and $\angle B=$$\angle E$ .

(b) In $\Delta ABC$ and $\Delta DEF$, $\angle B=$$\angle E=$ ${90}^0$, $AC=DF$ and $BC=EF$.

(c) In $\Delta ABC$ and $\Delta QRP$ , $AB=QR$, $\angle B=$$\angle R$ and $\angle C=$$\angle P$ .

(d) In $\Delta ABC$ and $\Delta PQR$ , $AB=PQ,AC=PR$ and $BC=QR$.

(e) In $\Delta ABC$ and $\Delta PQR$ , $BC=QR$, $\angle A=$ ${90}^{\circ }$, $\angle C=$ $\angle R=$ ${40}^{\circ }$ and $\angle Q=$${50}^{\circ }$.

Answer 1

(a) In $\Delta ABC$ and $\Delta DEF$,
$AB=DE$,
$BC=EF$
and $\angle B=$ $\angle E$ .
By SAS (Side Angle Side), the given triangles are congruent.

(b) In $\Delta ABC$ and $\Delta DEF$,
$\angle B=$ $\angle E=$ ${90}^{\circ }$
$AC=DF$ and $BC=EF$.
By RHS (Right angle Hypotenuse Side), the given triangles are congruent.

(c) In $\Delta ABC$ and$\Delta QRP$ ,
$AB=QR$,
$\angle B=$ $\angle R$
and $\angle C=$ $\angle P$.
By ASA(angle side angle), the given triangles are congruent.

(d) In $\Delta ABC$ and $\Delta PQR$,
$AB=PQ$,
$AC=PR$,
and $BC=QR$.
By SSS (side side side) the given triangles are congruent.

(e) In $\Delta ABC$ and $\Delta PQR$ ,
$BC=QR$,
$\angle A=$ ${90}^{\circ }$
$\angle C=$ $\angle$​​​​​​​$R={40}^{\circ }$​​​​​​​
and $\angle Q=$​​​​​​​ ${50}^{\circ }$​​​​​​​.
Sum of all angles $={180}^{\circ }$
$\Rightarrow P+Q+R=180$
$\Rightarrow P+40+50=180$
$\therefore P={90}^{\circ }$
$A=P$
$C=R$
$BC=QR$
By ASA (angle side angle), the given triangles are congruent.