Читать книгу Geometry for Students and Parents - Sergey D Skudaev - Страница 4

If two lines never intersect with each other, they are parallel. You can see the || symbol that is used to indicate parallel lines.

Per Fred́s theorem, if two parallel lines intersect a third line, two kinds of angles are formed: acute angles and obtuse angles. All acute angles are equal and all obtuse angles are equal. Adjacent angles add up to 180 degrees. See Figure 4.

Figure 4. AB || CD Angle 1 = 2 = 3 = 4. Angle 5 = 6 = 7 = 8

H is the point of the intercept of lines AB and EF. Similarly, G is the point of the intercept of lines CD and EF. Angles 1 and 5 are adjacent and add up to 180 degrees. Adjacent angles are: 1 and 6; 2 and 5; 6 and 2; 3 and 7; 3 and 8; 7 and 4; 8 and 4.

In geometry, angles are denoted by 3 letters starting with a letter that denotes any angle side. The angle 1 can be denoted by AHE or EHA.

The angle 5 can be denoted by EHB or BHE. The angle 2 can be denoted by BHG or GHB. The angle 6 can be denoted by AHG or GHA. The angle 8 can be denoted by CGF or FGC and so on.

There are three conditions that prove if two lines are parallel.

The first condition: If two lines are intersected by a third line and two inner angles adjacent to the third line add up to 180 degrees then the lines are parallel. See Figure 5.

Figure 5. If angle BGH + DHG =180 then AB || CD

The second condition: If two lines intersect the third line and the corresponding angles are equal, then these lines are parallel. See figure 6.

Figure 6. If angle BGE = DHG then AB || CD

The third condition: If two lines intersect the third line and the crosswise lying angles are equal then these two lines are parallel. See Figure 7

Figure 7. If angle AGH = DHG then AB || CD

If two lines intersect and form an angle of 90 degrees, they are perpendicular to each other.

In this case, all four angles are equal, and each angle is equal to 90 degrees. The _ | _ symbol is used to indicate that the lines are perpendicular.

AB _ | _ CD. See figure 8.

Figure 8. Perpendicular lines