Question

Triangle DEF is an isosceles triangle with DE = FE. if DE = 5 and /_ D = 75%,

Answer 1

First draw a triangle, with D at the top, E at bottom left and F at bottom right.

Dropping the perpendicular bisector from D, to length FE.
Angle D/2  = 75/2 = 37.5 degrees.
Let the bisector meet FE at G.
FE = FG + GE.

Sine 37.5 =  (GE)/5    Cross Multiplying.
GE  =  5Sin37.5.
But FE = Twice GE
FE =  2*(5Sin37.5) = 10Sin37.5  Evaluate with your calculator.
FE = 6.08

b) Measure of Angle F.
F is one of the base angles of the isosceles triangle and base angles are equal.

Let Angles of the Isosceles triangle be  x,  x,  75.

x + x + 75 = 180  Sum of angles in a triangle.

2x + 75 = 180.
2x = 180 -75
2x = 105.  Divide by 2.
x = 105/2  = 52.5

Therefore angle F = 52.5 degrees.

c)  Angle E = Angle F.
Angle E = 52.5 degrees.

Cheers.