Question

В четырёхугольнике ABCD известно, что AD = BC и AB = CD. Докажите, что углы B и D равны.

Answer 1

Если AB=BC, a AB=CD, то это параллелограм, поэтому у него противоположные углы равны. Исходя из свойств параллелограмма — углы B и D равны

Answer 2

Given Information

In the quadrilateral ABCD, it is known that AD = BC and AB = CD.

Proof

To prove that angles B and D are equal, we can use the fact that opposite sides of a parallelogram are equal in length and opposite angles are equal.

1. Given: AD = BC and AB = CD. 2. Since AD = BC, we can conclude that triangle ABD is congruent to triangle CBD by the Side-Side-Side (SSS) congruence criterion. 3. By congruence, we know that angle ABD is equal to angle CBD. 4. Similarly, since AB = CD, we can conclude that triangle ABC is congruent to triangle CDA by the SSS congruence criterion. 5. By congruence, we know that angle ABC is equal to angle CDA. 6. Now, consider the quadrilateral ABCD. Since angle ABD is equal to angle CBD (from step 3) and angle ABC is equal to angle CDA (from step 5), we can conclude that angle B is equal to angle D.

Therefore, we have proved that angles B and D are equal in the quadrilateral ABCD.