(a)(i): $326$
(a)(ii): reflex
(b): $c$, $d$
(c): $138$
(d): $73^\circ$
(e): $40$
(f): $23.6$ or $23.57$ to $23.58$

Detailed Answer

(a) (i)

(ii)
The angle \( k \) is reflex angle.

(b)
Alternate angle to \( x \): Angle \( c \)
Alternate angles (also called Z-angles) are equal and form a “Z” shape.

Corresponding angle to \( x \): Angle \( d \)
Corresponding angles are in the same relative position at each intersection (F-angles).

(c)
In a parallelogram, adjacent angles are supplementary (add up to \( 180^\circ \)).
$
y + 42^\circ = 180^\circ
$
$
y = 180^\circ – 42^\circ = 138^\circ
$
So, \( y = 138^\circ \).

(d)
According to the Central Angle Theorem, the measure of a central angle is twice the measure of an inscribed angle that subtends the same arc.
$\angle BOC = 2 \times \angle BAC = 2 \times 17^\circ = 34^\circ$.

In isosceles triangle BOC, $\angle OBC = \angle OCB$.
The sum of angles in triangle BOC is 180 degrees.
So, $\angle BOC + \angle OBC + \angle OCB = 180^\circ$.
 $34^\circ + \angle OBC + \angle OCB = 180^\circ$. 

$\angle OBC = \angle OCB$

$34^\circ + 2 \angle OCB = 180^\circ$.
$2 \angle OCB = 180^\circ – 34^\circ = 146^\circ$.
$\angle OCB = \frac{146^\circ}{2} = 73^\circ$.
$\angle ACB = 73^\circ$.

(e)

$
\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
$
$
171 = \frac{(n-2) \times 180}{n}
$
$
171n = 180(n – 2)
$
$
171n = 180n – 360
$
$
180n – 171n = 360
$
$
9n = 360
$
$
n = 40
$

(f)

$
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}
$

Opposite side = 7.4 cm
Hypotenuse = 18.5 cm
$
\sin(x) = \frac{7.4}{18.5}
$
$
\sin(x) = 0.4
$

Take the inverse sine
$
x = \sin^{-1}(0.4) \approx 23.58^\circ
$