You can use a point on a perpendicular bisector to prove that two segments are congruent. If the point is on the perpendicular bisector of a segment, then it’s equidistant from the endpoints of the segment. (Here’s an abbreviated version: If you have a perpendicular bisector, then there’s one pair of congruent segments.)
![image0.jpg](https://www.dummies.com/wp-content/uploads/271995.image0.jpg)
The above figure shows you how this equidistance theorem works.
![image1.png](https://www.dummies.com/wp-content/uploads/271996.image1.png)
Note that you can see the reasoning behind the short form of the theorem in the above diagram:
![image2.png](https://www.dummies.com/wp-content/uploads/271997.image2.png)
Here’s a proof that uses this equidistance theorem:
![image3.png](https://www.dummies.com/wp-content/uploads/271998.image3.png)
![image4.jpg](https://www.dummies.com/wp-content/uploads/271999.image4.jpg)
Statement 1:
![image5.png](https://www.dummies.com/wp-content/uploads/272000.image5.png)
Reason for statement 1: Given.
Statement 2:
![image6.png](https://www.dummies.com/wp-content/uploads/272001.image6.png)
Reason for statement 2: If sides, then angles.
Statement 3:
![image7.png](https://www.dummies.com/wp-content/uploads/272002.image7.png)
Reason for statement 3: Given.
Statement 4:
![image8.png](https://www.dummies.com/wp-content/uploads/272003.image8.png)
Reason for statement 4: If two congruent angles (angle 2 and angle 3) are added to two other congruent angles (angle 1 and angle 4), then the sums are congruent.
Statement 5:
![image9.png](https://www.dummies.com/wp-content/uploads/272004.image9.png)
Reason for statement 5: If angles, then sides.
Statement 6:
![image10.png](https://www.dummies.com/wp-content/uploads/272005.image10.png)
Reason for statement 6: If two points (M and Q) are equidistant from the endpoints of a segment (segment LN; see statements 1 and 5), then they determine the perpendicular bisector of that segment.
Statement 7:
![image11.png](https://www.dummies.com/wp-content/uploads/272006.image11.png)
Reason for statement 7: If a point (point P) is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.