Is it wrong to write ? Why?

Question

Homi · Accepted Answer

The triangles cannot be proven congruent because the conditions required for congruence are not met by the information given in the figure.

Step 1 — Identify common and given parts

Let us look at the two triangles, $\Delta\text{ABD}$ (green) and $\Delta\text{CBD}$ (red). They share a common side, which is BD. The diagram shows some angles are marked as equal: Angle A (in $\Delta\text{ABD}$ ) has one arc, and Angle C (in $\Delta\text{CBD}$ ) has one arc. So, $\angle\text{DAB} = \angle\text{BCD}$ . Angle ABD (in $\Delta\text{ABD}$ ) has two arcs, and Angle CBD (in $\Delta\text{CBD}$ ) has two arcs. So, $\angle\text{ABD} = \angle\text{CBD}$ .

Step 2 — Check congruence criteria

We need to check if any of the congruence rules (SSS, SAS, ASA, AAS, RHS) can be applied.

SSS (Side-Side-Side): This rule requires all three corresponding sides to be equal. We only know that BD is a common side. There is no information given that side AB is equal to side BC, or that side AD is equal to side CD. So, the SSS rule cannot be applied.
SAS (Side-Angle-Side): This rule requires two corresponding sides and the included angle between them to be equal. We know BD is a common side. We have some equal angles ( $\angle\text{DAB} = \angle\text{BCD}$ and $\angle\text{ABD} = \angle\text{CBD}$ ). However, we do not know if the sides around these equal angles are also equal. For example, we do not know if AB = BC or AD = CD. So, the SAS rule cannot be applied.
ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side): These rules require two corresponding angles and one corresponding side to be equal. We have some equal angles and a common side BD. However, the diagram also shows that Angle ADB (in $\Delta\text{ABD}$ ) has two arcs, while Angle CDB (in $\Delta\text{CBD}$ ) has one arc. This means $\angle\text{ADB} \neq \angle\text{CDB}$ . If the triangles were congruent, all three corresponding angles would have to be equal. This inconsistency in the angle markings means that ASA or AAS cannot be reliably applied.

Step 3 — Conclusion

Based on the analysis, we do not have enough consistent information to prove that $\Delta\text{ABD}$ is congruent to $\Delta\text{CBD}$ using any of the standard congruence rules.

Diagram 1

Answer

Yes, it is wrong to write $\Delta\text{ABD} \cong \Delta\text{CBD}$ according to the figure.

In the figure:

The two triangles ABD and CBD share the common side BD.

They also have some equal angles shown in the figure.

But for congruence, we need one of these conditions: SSS, SAS, ASA, AAS, or RHS.

Only one side (BD) is common. There is no information showing that AB = BC or AD = CD. So, the SSS congruence rule is missing.

The equal angle markings are not necessarily the included angle between known equal sides. We do not know that the sides around those angles are equal. So, the SAS congruence rule does not apply.

For ASA or AAS, while some angles are marked equal, the diagram shows that the third angles ( $\angle\text{ADB}$ and $\angle\text{CDB}$ ) are not equal. This means the angle information is inconsistent, so these rules cannot be applied.

Question 30

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