Geometric Twins | FIO

Question 15

Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent.

(a) $AB = DE$ , $BC = EF$ , $CA = DF$

(b) $AB = EF$ , $\angle A = \angle E$ , $AC = ED$

(d) $\angle A = \angle D$ , $\angle B = \angle E$ , $AC = DF$

(e) $AB = DF$ , $\angle B = \angle F$ , $AC = DE$

Solution

Understand the Question

Two triangles are congruent if all their corresponding sides and angles are equal. We can establish congruence using specific criteria:

SSS (Side-Side-Side): Three pairs of corresponding sides are equal.
SAS (Side-Angle-Side): Two pairs of corresponding sides and the included angle between them are equal.
AAS / ASA: Two pairs of angles and a corresponding side are equal.
RHS (Right angle-Hypotenuse-Side): In right-angled triangles, the hypotenuses and one pair of corresponding sides are equal.
SSA (Side-Side-Angle): Two sides and a non-included angle is not a valid congruence criterion.

(a) $AB = DE$ , $BC = EF$ , $CA = DF$

Step 1 · Check SSS Congruence

Diagram 1

Given: $AB = DE, \quad BC = EF, \quad CA = DF$

All three pairs of corresponding sides are equal.

By the SSS congruence criterion: $\triangle ABC \cong \triangle DEF$

Answer

(a) $\triangle ABC \cong \triangle DEF$ (by SSS criterion)

(b) $AB = EF$ , $\angle A = \angle E$ , $AC = ED$

Step 1 · Check SAS Congruence

Diagram 2

Given: $AB = EF, \quad \angle A = \angle E, \quad AC = ED$

Here, $\angle A$ is the included angle between $AB$ and $AC$ , and $\angle E$ is the included angle between $EF$ and $ED$ .

By the SAS congruence criterion: $\triangle ABC \cong \triangle EFD$

Answer

(b) $\triangle ABC \cong \triangle EFD$ (by SAS criterion)

(c) $AB = DF$ , $\angle B = \angle D = 90^\circ$ , $AC = FE$

Step 1 · Check RHS Congruence

Diagram 3

Given: $\angle B = \angle D = 90^\circ$ $\text{Hypotenuse } AC = \text{Hypotenuse } FE$ $\text{Side } AB = \text{Side } DF$

Both triangles are right-angled with equal hypotenuses and one pair of equal corresponding sides.

By the RHS congruence criterion: $\triangle ABC \cong \triangle FDE$

Answer

(c) $\triangle ABC \cong \triangle FDE$ (by RHS criterion)

(d) $\angle A = \angle D$ , $\angle B = \angle E$ , $AC = DF$

Step 1 · Check AAS Congruence

Diagram 4

Given: $\angle A = \angle D, \quad \angle B = \angle E, \quad AC = DF$

Two pairs of corresponding angles and one pair of non-included corresponding sides are equal.

By the AAS congruence criterion: $\triangle ABC \cong \triangle DEF$

Answer

(d) $\triangle ABC \cong \triangle DEF$ (by AAS criterion)

(e) $AB = DF$ , $\angle B = \angle F$ , $AC = DE$

Step 1 · Check SSA Condition

Diagram 5

Given: $AB = DF, \quad \angle B = \angle F, \quad AC = DE$

Here, $\angle B$ is not the included angle between $AB$ and $AC$ , and $\angle F$ is not the included angle between $DF$ and $DE$ .

Since SSA (Side-Side-Angle) is not a valid congruence rule, the triangles are not necessarily congruent.

Answer

(e) Not necessarily congruent (SSA is not a valid congruence criterion)

Common Mistakes

SSA Fallacy: Assuming any two sides and an angle guarantee congruence. The angle must be the included angle between the two sides for the SAS criterion to hold.
Incorrect Vertex Correspondence: Writing the triangle vertices in the wrong order. For example, in part (b), $\triangle ABC \cong \triangle EFD$ (since $A \leftrightarrow E$ , $B \leftrightarrow F$ , and $C \leftrightarrow D$ ), not $\triangle ABC \cong \triangle DEF$ .

Question 15

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