Question

Which of the following options gives the correct relation between the congruency of the figures given below?

• A. $△FGH\cong △IFJ\text{and}△AED\cong △ABC$
• B. $△GHF\cong △IJF\text{and}△ABC\cong △AED$
• C. $△HGF\cong △JFI\text{and}△BAC\cong △DEA$
• D. $△FGH\cong △FIJ\text{and}△CBA\cong △EDA$

Answer 1

The correct option is D $△FGH\cong △FIJ\text{and}△CBA\cong △EDA$

The illustrations of the transformations of the four triangles given above are shown below.
Considering $△ABC\text{and}△ADE:$


The above figure shows that $△ABC$ is flipped, i.e., has undergone reflection around a horizontal line parallel to the x-axis passing through $A(1,2)$ to get the image $△ADE$.

Considering $△FGH\text{and}△FIJ$:


The above figure shows that $△FGH$ is rotated counterclockwise by ${90}^{\circ }$ to get the image $△FIJ$ .

We know that rigid transformations like reflection and translation always produce congruent images.

Therefore, $△ABC\cong △ADE\text{and}△FGH\cong △FIJ.$ ------------------------ (Equation 1)
Let’s check all the options one-by-one to get the correct congruence relation among them.

In option A: $△FGH\cong △IFJ\text{and}△AED\cong △ABC$

Check if the corresponding sides of $△FGH\text{and}△IFJ$ are congruent.
In $△FGH\text{and}△IFJ,FG\cong IF,\text{but}GH$ is not congruent to $FJ$.
$\Rightarrow △FGH$ is not congruent to $△IFJ$.

In $△AED\text{and}△ABC,$
$AE$ is not congruent to $AB.$
$\Rightarrow △AED$ is not congruent to $△ABC$.

Thus, this is an incorrect congruence relation.

In option B: $△FGH\cong △FIJ\text{and}△CBA\cong △EDA$
From equation 1, we know that $△FGH\cong △FIJ$.
In $△CBA\text{and}△EDA,CB\cong ED,BA\cong DA,\text{and}CA\cong EA.$
This shows that all the corresponding sides are congruent; therefore, $△CBA\cong △EDA$ .
$\Rightarrow △FGH\cong △FIJ\text{and}△CBA\cong △EDA$
Thus, this is a correct congruence relation.

In option C: $△GHF\cong △IJF\text{and}△ABC\cong △AED$

Check if the corresponding sides of $△GHF\text{and}△IJF$ are congruent.

$GH\cong IJ,HF\cong JF,\text{and}GF\cong IF$

$\Rightarrow △GHF\cong △IJF$

In $△ABC\text{and}△AED,AB$ is not congruent to $AE.$
$\Rightarrow △ABC$ is not congruent to $△AED$
Thus, this is an incorrect congruence relation.

In option D: $△HGF\cong △JFI\text{and}△BAC\cong △DEA$

Check if the corresponding sides of $△HGF\text{and}△JFI$ are congruent.

$HG$ is not congruent to $JF$ .

$\Rightarrow △HGF$ is not congruent to $△JFI.$
In $△BAC\text{and}△DEA,$
$BA$ is not congruent to $DE.$

$\Rightarrow △BAC$ is not congruent to $△DEA$
Thus, this is an incorrect congruence relation.

➡Option B is correct.