Question

AB и CD-перпендикулярны к прямой BD, точки A и С лежат по разные стороны от прямой BD. Докажите , что BC параллелен AD, если AB=CD

Answer 1

ΔABD=ΔCBD ( по первому признаку ) → AD=BC
ABCD- параллелограмм ( противоположные стороны попарно равны)→ BC параллельна AD

Answer 2

Given Information:

We are given that line segments AB and CD are perpendicular to the line segment BD, and points A and C lie on opposite sides of the line segment BD. We need to prove that line segment BC is parallel to line segment AD, given that AB = CD.

Proof:

To prove that BC is parallel to AD, we can use the fact that if two lines are perpendicular to the same line, then they are parallel to each other.

Since AB and CD are perpendicular to BD, they are also perpendicular to each other. This is because if two lines are perpendicular to the same line, they are parallel to each other. Therefore, AB is parallel to CD.

Now, let's consider the triangle ABC. We know that AB is parallel to CD and AB = CD. By the converse of the corresponding angles theorem, we can conclude that angle BCA is equal to angle CAD. This is because corresponding angles formed by a transversal cutting two parallel lines are congruent.

Since angle BCA is equal to angle CAD, and angle BCD is a right angle (as BD is perpendicular to CD), we can conclude that angle BCD is also a right angle. This means that line segment BC is perpendicular to line segment CD.

Now, let's consider the triangle ACD. We know that angle BCD is a right angle and line segment BC is perpendicular to line segment CD. By the converse of the corresponding angles theorem, we can conclude that angle CAD is also a right angle. This means that line segment AD is perpendicular to line segment CD.

Since line segment AD is perpendicular to line segment CD, and line segment BC is perpendicular to line segment CD, we can conclude that line segment BC is parallel to line segment AD. This completes the proof.

Therefore, we have proved that if AB = CD and AB and CD are perpendicular to BD, then BC is parallel to AD.

Note: The proof provided above is based on the given information and the mathematical principles of parallel lines and perpendicular lines.