Question

Choose the correct criteria to prove that two triangles are congruent:
Criteria:

A) All three corresponding sides are congruent.
B) Two angles and the side between them are congruent.
C) Two angles and a non-included side are congruent .
D) Two sides and the angle between them are congruent.
For the given congruency theorem which option is suitable.
1) SSS (side-side-side) =
2) SAS (side-angle-side) =
3) AAS (angle-angle-side) =
4) ASA (angle-side-angle) ) =

Answer 1

1. SSS (Side-side-side)

All three corresponding sides are congruent.

Eg: Consider $△ABC$ and $△PQR$

$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}-\frac{1}{2}$

or $AB=PQ,BC=QR,AC=PR$

All the sides of triangle $ABC$ are equal to all sides of $△PQR$

2. SAS [Side-angle-side]

Two sides and the angle between them are equal

In $△ABC$ and $△PQR$,

$AB=PQ;BC=QR,\angle BAC=\angle QPR$

$\therefore △ABC\cong △PQR$ [by SAS Criteria]

3. AAS [angle-angle-side]

Two angles and a non- included sides are congruent.

Eg:

Consider the $△ABC$ and $△DEF$,

$BC||EF$ and $\angle ABC\cong \angle DEF$ .....(i)

$\angle BCA=\angle EFD$ [corresponding angles of parallel lines are congruent].....(ii)

$AD=CF$

$AD+CD=CF+CD$ [addition property of equality]

$AC=AD+CD$ [partition postulate]

$DF=CF+CD$ [partition postulate]

$AC=DF$ [substitution property]....(iii)

From (i), (ii) and (iii),

$△ABC\cong △DEF$ [AAS Criteria]

4. ASA [angle-side-angle]

Two angles and the side between them are congruent.

Eg:

$E$ is the mid-point of $NO$

$\angle SNW=\angle TOA$

Now, bisect $\angle SNE$ and $OA$ bisect $\angle TOE$ ....(i)

As $E$ is the midpoint, $NE=EO$.....(ii)

$\angle SEN\cong \angle TEO$ [vertical angles are congruent].....(iii)

From (i), (ii) and (iii),

$△SNE\cong △TOE$ [by ASA criteria]

i) SSS (side-side-side) = Option A
ii)SAS (side-angle-side) = Option D
iii)AAS (angle-angle-side) = Option C
iv) ASA (angle-side-angle) ) = Option B