Answer:

SSS: Here, we just aim to show that if we fix 3 side lengths of triangle ABC, then angles A, B, and C will be fixed. To do this, we use law of cosines, which states (we are letting  AB=c ,  AC=b , and  BC=a ):

a2+b2–2abcos(C)=c2  

In the equation, since we have fixed the side lengths, this equation will also fix  cos(C) , which will in turn tell us the measure of angle C (note that all angles must be less than  180∘ , so we only get one solution for angle C). We can repeat law of cosines for all 3 sides, which will give us values for angle A and angle B. Since all 3 angles are fixed once we have fixed the side lengths, any two triangles with the same 3 side lengths must be congruent.

SAS: We again fix two sides, just say AB and AC, as well as angle A. We can again perform law of cosines again, giving us

AB2+AC2–2(AB)(AC)cos(A)=BC2  

This gives us the value of BC, and then we will have 3 side lengths fixed. Once we have the 3 side lengths fixed, we can use SSS to fix the other two angles, and we will be done.

AAS and ASA: If we fix two angles in the triangle, then since the sum of the angles is 180, the third angle will also be fixed. Then, we can apply law of sines:

ABsin(C)=BCsin(A)=ACsin(B)  

We already know 3 angles, so we will know sin of all of those angles. Then, since we know one of the side lengths, say we already know AB, then we can just set  ABsin(C) equal to the other two fractions, and solve for side lengths BC and AC individually.

HL: I’m not sure what the P stands for in the question, but this case is when you know the hypotenuse and the leg of a right triangle. In this case, we can use Pythagorean Theorem to find the other leg, which reduces it to SSS.

Note on why SSA does not guarantee congruent triangles

Step-by-step explanation: