Question

Доказательство перпендикулярные прямые свойства​

Answer 1

Якщо одна з двох паралельних прямих перпендикулярна до третьої прямої, то й інша перпендикулярна до цієї прямої. Пряма, що перетинає площину, називається перпендикулярною цій площині, якщо вона перпендикулярна кожній прямій, яка лежить у цій площині. Перпендикулярність прямої і площини позначається як a ⊥ α .

Answer 2

Perpendicular Lines: Properties and Proof

Perpendicular lines are a fundamental concept in geometry. When two lines are perpendicular, they intersect at a right angle, forming a 90-degree angle. Here are some properties and proofs related to perpendicular lines:

1. Equal Lengths: If two lines are perpendicular, the line segments they form between the point of intersection and any point on the lines are equal in length. This means that if AB, BC, CD, and AD are line segments formed by the intersection of four perpendicular lines, then AB = BC = CD = AD.

2. Equal Angles: When two lines are perpendicular, the angles formed by the intersection are equal. In other words, if A and C are angles formed by the intersection of two perpendicular lines, and B and D are the other two angles formed, then A = C and B = D.

3. Opposite Sides: If two lines are perpendicular, the line segments connecting opposite sides of the intersection point are equal in length. For example, if BO = OD and AO = OC, where O is the intersection point of two perpendicular lines, then BO = OD and AO = OC.

4. Supplementary Angles: The sum of the angles formed by the intersection of two perpendicular lines is 180 degrees. In other words, if A and D are angles formed by the intersection of two perpendicular lines, then A + D = 180 degrees.

5. Diagonals: In a rectangle or square, the diagonals are perpendicular to each other. This property holds true for all rectangles and squares, regardless of their size or orientation.

6. Orthogonal Coordinate System: In an orthogonal coordinate system, the x-axis and y-axis are perpendicular to each other. This property is fundamental in graphing and analyzing functions in two dimensions.

7. Right Triangles: In a right triangle, the sides that form the right angle are perpendicular to each other. This property is one of the defining characteristics of a right triangle.

These properties and proofs provide a foundation for understanding and working with perpendicular lines in geometry. They are essential in various mathematical applications, such as calculating distances, determining angles, and solving geometric problems.