We have ∠D, ∠E, and ∠F and side [tex]d[/tex], side [tex]e[/tex], and side [tex]f[/tex] as shown in the diagram below
by Cosine rule, we work out the angle D
[tex] d^{2}= e^{2} + f^{2} -2(e)(f)cos(D)[/tex]
[tex] 3^{2}= 12^{2}+ 12^{2} -2(12)(12)cos(D)[/tex]
[tex]9=144+144-288cos(D)[/tex]
[tex]9=288-(288cos(D))[/tex]
[tex]9-288=-288cos(D)[/tex]
[tex]-279=-288cos(D)[/tex]
[tex] \frac{-279}{-288}=cos(D) [/tex]
[tex]D= cos^{-1} ( \frac{279}{288}) [/tex]
[tex]D=14.36[/tex] rounded to 2 dp
Triangle DEF is an isosceles triangle, so we can work out ∠E and ∠F by using the angles in triangle rules
180°-14.36 = 165.64°
165.64°÷2=82.9° rounded to 1 dp
so now we have
∠D = 14.36
∠E = ∠F = 82.9
Smallest angle is angle D
