Is it wrong to write ΔBAD ≅ ΔCDB? Why?

Question

Homi · Accepted Answer

When we say two triangles are congruent, it means they are exactly the same shape and size. The order of the letters in the congruence statement tells us which parts of the triangles match up.

Step 1 — Identifying the Triangles and Properties of a Rectangle

Let us look at the rectangle ABCD shown in the diagram. In a rectangle, all four angles are right angles (90 degrees). Also, the opposite sides of a rectangle are equal in length. So, we know that AB = CD and AD = BC.

The diagonal BD divides the rectangle into two triangles: $\Delta BAD$ (shaded red) and $\Delta CDB$ (shaded green). We need to check if the statement $\Delta BAD \cong \Delta CDB$ is correct.

Diagram 1

Step 2 — Checking the Vertex Correspondence for Congruence

The statement $\Delta BAD \cong \Delta CDB$ implies a specific matching of vertices:

The first vertex B of $\Delta BAD$ corresponds to the first vertex C of $\Delta CDB$ .
The second vertex A of $\Delta BAD$ corresponds to the second vertex D of $\Delta CDB$ .
The third vertex D of $\Delta BAD$ corresponds to the third vertex B of $\Delta CDB$ .

Now, let us check if the corresponding sides are equal based on this specific order:

Side BA (from $\Delta BAD$ ) should be equal to side CD (from $\Delta CDB$ ). We know that AB and CD are opposite sides of the rectangle. So, BA = CD. This part matches.
Side AD (from $\Delta BAD$ ) should be equal to side DB (from $\Delta CDB$ ). AD is a side of the rectangle. DB is the diagonal of the rectangle. A side of a rectangle is generally not equal to its diagonal. So, AD $\neq$ DB. This part does not match.
Side BD (from $\Delta BAD$ ) should be equal to side CB (from $\Delta CDB$ ). BD is the diagonal of the rectangle. CB is a side of the rectangle. Again, a diagonal is generally not equal to a side. So, BD $\neq$ CB. This part also does not match.

Since the corresponding sides are not equal when we follow the specific order of vertices given in $\Delta BAD \cong \Delta CDB$ , the triangles are not congruent under this particular correspondence.

Answer

Yes, it is wrong to write $\Delta BAD \cong \Delta CDB$ in a rectangle ABCD.

The reason is that the order of vertices in a congruence statement is very important. It tells us which parts of the triangles match up. If we try to match $\Delta BAD$ with $\Delta CDB$ in that specific order:

Side BA corresponds to side CD. These are equal (opposite sides of a rectangle).
Side AD corresponds to side DB. These are not equal (a side is not equal to a diagonal).
Side BD corresponds to side CB. These are not equal (a diagonal is not equal to a side).

Since the corresponding parts are not equal for this specific vertex matching, the statement $\Delta BAD \cong \Delta CDB$ is incorrect. However, the two triangles themselves are congruent, but with a different vertex correspondence, such as $\Delta BAD \cong \Delta DCB$ .

Question 15

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