In geometry, a trapezoid is a quadrilateral with at least one pair of parallel sides. In this particular case, we are given the trapezoid ABCD, where AB is parallel to CD. Our goal is to prove that line segment BD is equal in length to line segment CA.

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## Proof:

Let’s start by looking at the given trapezoid ABCD:

**ABCD is a trapezoid with AB || CD**- Let E be the intersection point of AD and BC
- Let F be the intersection point of the diagonals AC and BD

### Proof by Similar Triangles:

Now, let’s consider triangles AED and CEB:

**Triangle AED is similar to triangle CEB****Angle AED = Angle CEB (corresponding angles)****Angle ADE = Angle CBE (alternate interior angles)****Angle EAD = Angle EBC (corresponding angles)**

Therefore, triangles AED and CEB are similar by the Angle-Angle (AA) similarity criterion.

By the property of similar triangles, we have:

AE/CE = ED/EB

Now, let’s consider triangles AFE and CFD:

**Triangle AFE is similar to triangle CFD****Angle AFE = Angle CFD (corresponding angles)****Angle AEF = Angle CDF (alternate interior angles)****Angle FAE = Angle FCD (corresponding angles)**

Therefore, triangles AFE and CFD are similar by the Angle-Angle (AA) similarity criterion.

By the property of similar triangles, we have:

AF/CF = EF/FD

Now, let’s consider triangles AEF and CDF:

**Triangle AEF is similar to triangle CDF****Angle AEF = Angle CDF (corresponding angles)****Angle EAF = Angle DCF (alternate interior angles)****Angle AFE = Angle CFD (corresponding angles)**

Therefore, triangles AEF and CDF are similar by the Angle-Angle (AA) similarity criterion.

By the property of similar triangles, we have:

AE/CD = EF/FD

Substitute AE/CE = ED/EB into AE/CD = EF/FD:

ED/EB = EF/FD

Cross-multiply to get:

ED * FD = EB * EF

By the converse of the Thales’ theorem, we know that if two lines are cut proportionally by a transversal, then they are parallel. Therefore, line BD is parallel to line CA.

Since we have proven that BD is parallel to CA, and AB is parallel to CD, we have a pair of parallel lines. By the Converse of the Alternate Interior Angles Theorem, we know that BD is congruent to CA.

Therefore, we have proved that in trapezoid ABCD with base angles **AB || CD**, the segments **BD** and **CA** are equal in length.

### Conclusion:

By using the properties of similar triangles and the Converse of the Alternate Interior Angles Theorem, we have successfully proved that in trapezoid ABCD with base angles **AB || CD**, the segments **BD** and **CA** are equal in length.