Objectives
 Explore the right triangle congruence shortcut theorems — HL, LL, HA, and LA.
 Discover the relationship between right triangle congruence theorems and the congruence theorems for nonright triangles.
 Use the perpendicular bisector theorem to find unknown side lengths or determine if two right triangles are congruent.
 Use the angle bisector theorem to find unknown angle measures in right triangles.
Key Terms
 HA Congruence Theorem – “Hypotenuse – (acute) Angle”
 A theorem stating that if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the right triangles are congruent.
 Special case of AAS – right angle, acute angle, hypotenuse
 HL congruence theorem – “Hypotenuse – Leg”
 A theorem stating that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the right triangles are congruent.
 LA Congruence Theorem – “Leg – (acute) Angle”
 A theorem stating that if the leg and an acute angle of one right triangle are congruent to the leg and a corresponding acute angle of another right triangle, then the triangles are congruent.
 Special case of ASA – acute angle, leg, right angle or right angle, leg, acute angle
 Leg is included between angles.
 Special case of AAS – acute angle, right angle, leg
 Leg is not included between angles.
 LL Congruence Theorem – “Leg – Leg”
 A theorem stating that if the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the right triangles are congruent.
 Special case of SSS – Leg, leg, hypotenuse (use the Pythagorean theorem to find the missing hypotenuse, resulting in knowing the lengths and congruence of all 3 sides).
 Special case of SAS – Leg, right angle, leg (with the right angle included between the legs of these triangles).
Notes
Congruence Shortcuts for Right Triangles 

Examples 


Proving Theorems with Congruent Right Triangles 


