Five rays drawn from the same point divide the full angle in 5 congruent parts. How many different angles do these 5 rays form?

Does different mean non congruent? Then it is 5. It not then it is 25.

Figures can be different, yet congruent. So, assuming that you included full angles (but excluded zero angles, or angles greater than full ones), 25 seems a correct answer, and indeed there are 5 congruence types among them.

I didn't consider zero angles or angles greater then full ones, but now since you have brought them to my attention, I see no reoson to exclude them. Can we say there are infinite angles, each being equal 72k where k is a whole number?

Exercise 28: *Is the sum of the angles $14 ^{\circ} 24' 14"$ and $75 ^{\circ} 35' 25"$ acute or obtuse?*

$75 ^{\circ} + 14 ^{\circ} = 89 ^{\circ}$

$24' + 35' = 59'$

$44" + 25" = 69"$

$90 ^{\circ} 9"$ is obtuse.

Exercise 30: *Can both angles, whose sum is the straight angle, be acute? Obtuse?*

Both cannot be Acute: $(<90^{\circ} + <90^{\circ}) <180^{\circ}$

Both cannot be Obtuse: $(>90^{\circ} + >90^{\circ}) >180^{\circ}$

Exercise 33: *One of the angles formed by two intersecting lines is $\frac{2d}{5}$. Find the measure of the other three.*

$\frac{2d}{5} = \frac{2(90^{\circ})}{5} = \frac{180^{\circ}}{5} = 72^{\circ}$

$180^{\circ} - 72^{\circ} = 108^{\circ}$

$108^{\circ}$, $72^{\circ}$, $108^{\circ}$

Exercise 34: *Find the measure of an angle which is congruent to twice its supplementary one.*

$x + y = 180^{\circ}$

$x - 2y = 0^{\circ}$

$\frac{180^{\circ}}{3} = 60^{\circ}$

$2(60^{\circ}) = 120^{\circ}$

The angle measures: $120^{\circ}$

Exercise 35: //Two angles $ABC$ and $CBD$ having the common vertex $B$ and the common side $BC$ are positioned in such a way that they do not cover one another. The angle $ABC$ = $100^{\circ} 20'$, and the angle $CBD = 79^{\circ} 40'$. Do the sides $AB$ and $BD$ form a straight line or a bent one?

$100^{\circ} 20' + 79^{\circ} 40' = 180^{\circ}$

Straight line, $\S 22$ supplementary angles, the non-common sides form a straight line.

Exercise 36: *Two distinct rays, perpendicular to a given line, are erected at a given point. Find the measure of the angle between these rays.*

The rays being distinct must be on opposite sides of the line. Then they continue one another and are congruent to the straight angle, $180^{\circ}$.

Exercise 37: *In the interior of an obtuse angle, two perpendiculars to its sides are erected at the vertex. Find the measure of the obtuse angle, if the angle between the perpendiculars is $\frac{4d}{5}$.*

$\frac{4d}{5} = \frac{4(90^{\circ})}{5} = \frac{360^{\circ}}{5} = 72^{\circ}$

$2(90^{\circ} - 72^{\circ}) = 2(18^{\circ}) = 36^{\circ}$ in the remaining area of the obtuse angle.

Obtuse angle measures: $72^{\circ} + 36^{\circ} = 108^{\circ}$