Explanation:
trapezium means especially
AD || BC
AC = BC tells us that ABC is an isoceles triangle (both legs are identically long, which makes also their angles with their baseline AB equal).
therefore, A1 = B = 80°.
and because the sum of all angles in a triangle is always 180°, we get
180 = 80 + 80 + C1
C1 = 20°
(e)(1)
because of the law of a line intersecting parallel lines with the same angles, and AC is intersecting the parallel lines AD and BC, we know that
C1 = A2 = 20°
(e)(2)
because also the sum of all angles around a single point on one side of a line is 180°, we know D1 + D2 = 180°.
we also know that D2 is an angle in an equilateral triangle (that means all 3 sides are equally long, and therefore all 3 angles are equal too = 180/3 = 60°) and therefore
D2 = 60°.
therefore, D1 = 180 - 60 = 120°.
so, for the triangle ACD we have
180 = A2 + D1 + C2 = 20 + 120 + C2
C2 = 40° = 2×20 = 2×C1
(f)(1)
a kite is a quadrilateral, and as such the sum of all angles is 360°.
again remember the symmetry of a kite, so the angles at H and at F must be equal.
H = F = x
so, we have
360 = 110 + 50 + H + F = 110 + 50 + 2x
200 = 2x
x = 100°
(f)(2)
after drawing EG we see that this line splits the kite into 2 equal (ahem, congruent) triangles due to the symmetry of a kite exactly along that line.
and that splits both angles E and G in half for each of the triangles.
again, the sum of all angles in a triangle is 180°, so
180 = 110/2 + 50/2 + x = 55 + 25 + x = 80 + x
x = 100°