Question

в четырех угольнике ABCD все стороны равны. Докажите,что противолежащие углы A и C равны. ( Подсказка: сделайте дополнительное построение: проведите диагональ BD.)

Answer 1

1)Построим диагональ ВD,она разделит АВСD на 2 треугольника.
2)Рассмотрим треугольник DAB и треугольник DBC,у них:
DВ-общая
АВ=ВС(по условию)
АD=DС(по условию);зн.треугольник DAB=DBC(по трем сторонам);след.угол А=углу С(по свойству равных треугольников)ч.т.д.

Answer 2

Proof that Opposite Angles A and C are Equal in a Quadrilateral ABCD

To prove that the opposite angles A and C in quadrilateral ABCD are equal, we can use the fact that all sides of the quadrilateral are equal and then make use of the additional construction of the diagonal BD.

1. Given: - Quadrilateral ABCD with all sides equal (AB = BC = CD = DA).

2. To Prove: - ∠A = ∠C.

3. Proof: - Since all sides of the quadrilateral are equal, we can consider it as a rhombus. - Let's consider the additional construction of the diagonal BD.

4. Additional Construction: - We draw the diagonal BD.

5. Observation: - In a rhombus, the diagonals bisect each other at right angles. This means that ∠ABD = ∠CBD and ∠BAD = ∠BCD.

6. Conclusion: - From the observation, we can conclude that ∠A = ∠C.

Therefore, we have proved that the opposite angles A and C in quadrilateral ABCD are equal.