Answer:
The measures of the triangle are \(30\°,60\°,90\°\)
Step-by-step explanation:
see the attached figure with letters to better understand the problem
Step 1
Verify if the triangle ABC is a right triangle
Applying the Pythagoras Theorem
\((14\sqrt{3})^{2} =21^{2}+(7\sqrt{3})^{2}\\ \\588=441+147\\ \\588=588\)
Is a right triangle
therefore
the measure of angle A is \(90\°\)
Step 2
In the right triangle ABC
\(cos(C)=\frac{21}{14\sqrt{3}}\)
\(C=arcos(\frac{21}{14\sqrt{3}})=30\°\)
Step 3
Find the measure of angle B
we know that
the sum of the internal angles of a triangle is equal to \(180\°\)
so
\(A+B+C=180\°\)
substitute the values and solve for B
\(90\°+B+30\°=180\°\)
\(B+120\°=180\°\)
\(B=60\°\)
The measures of the triangle are \(30\°,60\°,90\°\)
