The HLR (Hypotenuse-Leg-Right angle) theorem — often called the HL theorem — states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. The following figure shows you an example.
![image0.jpg](https://www.dummies.com/wp-content/uploads/269983.image0.jpg)
You can call this theorem HLR (instead of HL) because its three letters emphasize that before you can use it in a proof, you need to have three things in the statement column (congruent hypotenuses, congruent legs, and right angles).
Note: When you use HLR, listing the pair of right angles in a proof statement is sufficient for that part of the theorem; you don’t need to state that the two right angles are congruent.
Ready for an HLR proof? Well, ready or not, here you go.
![image1.png](https://www.dummies.com/wp-content/uploads/269984.image1.png)
![image2.jpg](https://www.dummies.com/wp-content/uploads/269985.image2.jpg)
Here’s a possible game plan. You see the pair of congruent triangles and then ask yourself how you can prove them congruent. You know you have a pair of congruent sides because the triangle is isosceles.
![image3.png](https://www.dummies.com/wp-content/uploads/269986.image3.png)
Here’s the proof:
Statement 1:
![image4.png](https://www.dummies.com/wp-content/uploads/269987.image4.png)
Reason for statement 1: Given.
Statement 2:
![image5.png](https://www.dummies.com/wp-content/uploads/269988.image5.png)
Reason for statement 2: Definition of isosceles triangle.
Statement 3:
![image6.png](https://www.dummies.com/wp-content/uploads/269989.image6.png)
Reason for statement 3: Reflexive Property.
Statement 4:
![image7.png](https://www.dummies.com/wp-content/uploads/269990.image7.png)
Reason for statement 4: Given.
Statement 5:
![image8.png](https://www.dummies.com/wp-content/uploads/269991.image8.png)
Reason for statement 5: Definition of altitude.
Statement 6:
![image9.png](https://www.dummies.com/wp-content/uploads/269992.image9.png)
Reason for statement 6: Definition of perpendicular.
Statement 7:
![image10.png](https://www.dummies.com/wp-content/uploads/269993.image10.png)
Reason for statement 7: HLR (using lines 2, 3, and 6)
Statement 8:
![image11.png](https://www.dummies.com/wp-content/uploads/269994.image11.png)
Reason for statement 8: CPCTC.
Statement 9:
![image12.png](https://www.dummies.com/wp-content/uploads/269995.image12.png)
Reason for statement 9: Definition of midpoint.
Statement 10:
![image13.png](https://www.dummies.com/wp-content/uploads/269996.image13.png)
Reason for statement 10: Definition of median.