In right triangles the right angles are always equal, so the congruence criteria become simpler. Leg-leg: if two legs of one triangle equal two legs of another, the triangles are equal (from SAS). Leg-angle: if a leg and an adjacent acute angle are equal, the triangles are equal (from ASA). Hypotenuse-angle: if the hypotenuse and an acute angle are equal, the triangles are equal, since the other acute angles are also equal and ASA applies. Hypotenuse-leg: if the hypotenuse and one leg are equal, the triangles are equal; in the proof the triangles are put together along the equal legs to form an isosceles triangle; its altitude to the base is also a bisector, so the angles at the common leg are equal and SAS applies.
Worked examples
△ABC and △DEF are right triangles (C and F right angles), AB = DE = 13 cm, BC = EF = 5 cm — equal by hypotenuse-leg, so AC = DF.
Two right triangles have equal hypotenuses and an acute angle of 35° each — equal by hypotenuse-angle; the other acute angles are 55° each.
Class activity
“Criterion cards”: pupils look at pairs of right triangles and name the criterion: LL, LA, HA or HL.
Practice
1
Right triangles △ABC and △MNK have legs 6, 8 and 6, 8. By which criterion are they equal?
Leg-leg
2
The hypotenuses are equal and one leg is equal. Which criterion?
Hypotenuse-leg
3
Two right triangles have hypotenuse 10 and an acute angle of 40°. What is the other acute angle (degrees)?
50
4
Why are two equal elements enough for right triangles instead of three?
The right angles are always equal, which is one equality given in advance.