If two parallel lines are intersected by a transversal, prove that the bisectors of the interior angles on the same side of the transversal intersect each other at right angles.
To prove that the bisectors of the interior angles on the same side of a transversal intersect each other at right angles, we can use the concept of parallel lines and the properties of angles.
Let's consider two parallel lines, line m and line n, which are intersected by a transversal line t. We will label the points of intersection as A, B, C, and D, as shown below:
```
A --------- B
| |
| |
| t |
| |
| |
C --------- D
```
From this diagram, we have two pairs of corresponding angles:
1) Angle A and angle C on the left side of the transversal.
2) Angle B and angle D on the right side of the transversal.
Since line m and line n are parallel, the corresponding angles are equal:
1) Angle A = angle C.
2) Angle B = angle D.
Now, let's draw the bisectors of these angles.
For angle A, draw the bisector, which is a line that divides the angle into two equal parts. Let's label the point of intersection with line m as E.
For angle C, draw the bisector, which is a line that divides the angle into two equal parts. Let's label the point of intersection with line m as F.
For angle B, draw the bisector, which is a line that divides the angle into two equal parts. Let's label the point of intersection with line n as G.
For angle D, draw the bisector, which is a line that divides the angle into two equal parts. Let's label the point of intersection with line n as H.
From the diagram, we have the following:
1) Angle A = angle C (Given)
2) Angle B = angle D (Given)
3) Angle A + angle B = 180 degrees (Angles on a straight line)
4) Angle C + angle D = 180 degrees (Angles on a straight line)
5) Angle A + angle B = angle C + angle D (Combining equations 3 and 4)
Now, we need to prove that the bisectors of angle A and angle B intersect at a right angle.
To do this, we will use contradiction. If the bisectors of angle A and angle B do not intersect at a right angle, it means they intersect at some other angle, let's say X.
Assume that angle X is not 90 degrees. This means that angle AEX and angle BGX are not 90 degrees.
Since angle AEX and angle BGX are not 90 degrees, their sum is less than 180 degrees. Therefore, angle AEX + angle BGX < 180 degrees.
But, we know that angle A + angle B = 180 degrees. Since angle AEX and angle BGX are the bisectors of angle A and angle B respectively, we can write:
angle AEX + angle BGX = angle A + angle B
From equation 5, we have:
angle A + angle B = angle C + angle D
Comparing the two equations:
angle AEX + angle BGX = angle A + angle B = angle C + angle D
This shows that angle AEX + angle BGX = angle C + angle D
Therefore, angle AEX + angle BGX > angle AEX + angle BGX, which contradicts the assumption that angle X is not 90 degrees.
Hence, our assumption is wrong and the bisectors of angle A and angle B intersect at a right angle.
Therefore, we have proved that if two parallel lines are intersected by a transversal, the bisectors of the interior angles on the same side of the transversal do intersect each other at right angles.
consider the fact the two angles add to 180°
so, what is the sum of their bisectors?