To find the measure of ∠a in ∆abc, use theorem, law of sines, law of cosines). to find the length of side hi in ∆hig, use theorem, law of sines, law of cosines) .

Part 1) To find the measure of ∠A in ∆ABC, use

we know that

In the triangle ABC

Applying the law of sines

in this problem we have

therefore

the answer  Part 1) is

Law of Sines

Part 2) To find the length of side HI in ∆HIG, use

we know that

In the triangle HIG

Applying the law of cosines

In this problem we have

g=HI

G=angle Beta

substitute

therefore

the answer Part 2) is

Law of Cosines

for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis: