Question

В окружности проведены диаметры DF и HK. Докажите, что прямые DH и FK параллельны.

Answer 1

В двух желтых треугольниках углы при вершине О равны, т.к. они вертикальные, стороны равны, т.к. это радиусы одной окружности. Следовательно, эти треугольники равны, равны и соответственные углы в них: ∠Н=∠К. Углы Н и К являются накрест лежащими при прямых DH и FK, значит, по соответствующему признаку DH||FK

Answer 2

Proving the Parallelism of DH and FK

To prove that the lines DH and FK are parallel, we can use the property that if a quadrilateral's opposite sides are equal, then its opposite angles are equal, and the quadrilateral is a parallelogram.

Given: - In a circle, diameters DF and HK are drawn.

To Prove: - DH and FK are parallel.

Proof: 1. In a circle, any diameter subtends a right angle at any point on the circle. - "In a circle, any diameter subtends a right angle at any point on the circle".

2. Since DF and HK are diameters, they subtend right angles at any point on the circle. - "DF and HK are diameters, they subtend right angles at any point on the circle".

3. Consider the quadrilateral DHKF. - "Consider the quadrilateral DHKF".

4. DF and HK are diameters, so DK is a straight line passing through the center of the circle. - "DF and HK are diameters, so DK is a straight line passing through the center of the circle".

5. DK is a straight line, so angle DHK and angle DFK are right angles. - "DK is a straight line, so angle DHK and angle DFK are right angles".

6. Since angle DHK and angle DFK are right angles, DHKF is a parallelogram. - "Since angle DHK and angle DFK are right angles, DHKF is a parallelogram".

7. In a parallelogram, opposite sides are equal. - "In a parallelogram, opposite sides are equal".

8. Therefore, DH and FK are parallel. - "Therefore, DH and FK are parallel".

Hence, it is proved that the lines DH and FK are parallel.