From DIAGRAM 1, i. If, f = 60° and c = 8x – 4, find the value of x ii. If e = 110 find the value of a,b,c,d,f,g and h

From the diagram, angles $c$ and $f$ are alternate interior angles. Since lines AB and CD are parallel (implied by the context of the problem), alternate interior angles are equal.

$$c = f$$

$$8x = 60 + 4$$ $$8x = 64$$ $$x = \frac{64}{8}$$ $$x = \boxed{8}$$

• Angles $e$ and $h$ are vertically opposite angles, so they are equal.

$$h = e = 110^\circ$$

• Angles $e$ and $f$ form a linear pair (angles on a straight line), so their sum is $180^\circ$.

$$e + f = 180^\circ$$ $$110^\circ + f = 180^\circ$$ $$f = 180^\circ - 110^\circ = 70^\circ$$

• Angles $f$ and $g$ are vertically opposite angles, so they are equal.

$$g = f = 70^\circ$$

So, $e = 110^\circ$, $f = 70^\circ$, $g = 70^\circ$, $h = 110^\circ$.

Since lines AB and CD are parallel:

• Angles $a$ and $e$ are corresponding angles, so they are equal.

$$a = e = 110^\circ$$

• Angles $b$ and $f$ are corresponding angles, so they are equal.

$$b = f = 70^\circ$$

• Angles $c$ and $g$ are corresponding angles, so they are equal.

$$c = g = 70^\circ$$

• Angles $d$ and $h$ are corresponding angles, so they are equal.

$$d = h = 110^\circ$$

$$a = \boxed{110^\circ}$$ $$b = \boxed{70^\circ}$$ $$c = \boxed{70^\circ}$$ $$d = \boxed{110^\circ}$$ $$f = \boxed{70^\circ}$$ $$g = \boxed{70^\circ}$$ $$h = \boxed{110^\circ}$$