ABCD is a rhombus. Explain why Triangle A B C is congruent to triangle C D A.

Triangle A B C is congruent to triangle C D A because all four sides of a rhombus are equal in length. This means that the two triangles have the same side lengths, and therefore they are congruent.

To understand why Triangle ABC is congruent to Triangle CDA, we need to look at the properties of a rhombus.

A rhombus is a quadrilateral with four equal sides. In other words, all sides of a rhombus have the same length. Additionally, the opposite angles in a rhombus are congruent, which means they have the same measure.

Now, let's examine Triangle ABC. It is a triangle formed by three vertices, namely, A, B, and C. Since ABCD is a rhombus, we know that AB = BC (because adjacent sides of a rhombus are equal in length) and angle ABC = angle BCA (because opposite angles of a rhombus are congruent).

If we consider Triangle CDA, we can see that CD = DA (because opposite sides of a rhombus are equal in length) and angle CDA = angle DAC (because opposite angles of a rhombus are congruent).

By comparing the corresponding sides and angles of Triangle ABC and Triangle CDA, we find that AB = CD, BC = DA, and angle ABC = angle CDA.

According to the definition of congruent triangles, if all corresponding sides and angles of two triangles are equal, then the triangles are congruent. Therefore, Triangle ABC is congruent to Triangle CDA.

In conclusion, Triangle ABC is congruent to Triangle CDA because they have the same side lengths and angle measures, which can be deduced from the properties of a rhombus.

To prove that triangle ABC is congruent to triangle CDA, we need to show that their corresponding sides and angles are equal.

In a rhombus, all sides are equal in length. Therefore, sides AB and CD are equal.

Additionally, opposite angles in a rhombus are congruent. So angle ABC is equal to angle CDA.

We also know that adjacent angles in a rhombus are supplementary, meaning they add up to 180 degrees. So angle BCD and angle CAD are supplementary.

Using the Side-Angle-Side (SAS) congruence criterion, we have two pairs of congruent sides and a pair of congruent angles. Therefore, triangle ABC is congruent to triangle CDA.

To summarize:
- Side AB is equal to side CD (both are sides of the rhombus).
- Angle ABC is equal to angle CDA (opposite angles in a rhombus are congruent).
- Angle BCD is equal to angle CAD (adjacent angles in a rhombus are supplementary).

Hence, triangle ABC is congruent to triangle CDA.