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Based on the diagram above, the following relation holds true.
If ∠ A≅ ∠ B or ∠ C ≅ ∠ D, then the trapezoid ABCD is isosceles.
Consider an auxiliary line that passes through C and is parallel to AD. Let E be the intersection point of this line and AB.
Applying the Parallelogram Opposite Sides Theorem, it can be concluded that AECD is a parallelogram. Consequently, the opposite sides are congruent. AD ≅ EC Additionally, ∠ BEC and ∠ A are congruent because of the Corresponding Angles Theorem. Since ∠ A and ∠ B are also congruent angles, by the Transitive Property of Congruence, ∠ BEC is congruent to ∠ B. ∠ BEC ≅ ∠ A ∠ A ≅ ∠ B ⇒ ∠ BEC ≅ ∠ B Therefore, ∠ BEC can be marked using one angle marker.
By the Converse Isosceles Triangle Theorem, △ BEC is isosceles. Therefore, EC and BC are congruent. Lastly, using the Transitive Property of Congruence once again, it is obtained that AD≅BC. Consequently, ABCD is an isosceles trapezoid.
