Matt Soren writes: "I am having trouble with construction exercise #27. I cannot figure out how, only using a compass, I can draw a one degree arc, given a 19 degree arc."

Dear Matt,

As a general hint, one can suggest: If a very special question causes a difficulty, one can try to ask a more general question. Making progress in a more general question often turns out easier than attacking a more special one head-on.

In this problem, OK, you don't know how to construct a $1^{\circ}$ arc from a $19^{\circ}$ arc. Here is a more general question: Find degree measures of all those arcs that can be constructed (to begin - *you can construct*) from a $19^{\circ}$ arc on a given circle, using the compass.

Alexander

Hello,

I have solved Exercise 27 in my head, but on paper it never works out correctly. In other words, I've proved that the construction I have in mind should work, but when I draw it out, it doesn't. I feel like this is just an inherent limitation in the precision of my construction, which is highly susceptible to drift due to the fact that many steps are involved, and with each step there is a human error factor associated with perhaps placing the compass a fraction of a degree away from the precisely correct location.

Perhaps there is a better construction that is not so susceptible to drift, but the construction I am using has me repeating the 19 degree angle many times, and I feel like with each repetition of the 19 degree angle there is a drift of less than 1 degree, but when added up and I finally achieve my "1 degree" angle, it has drifted so much that it is actually more like an 8 or 9 degree angle.

Do you have any advice for how to make more accurate constructions?

Exercise 21: *In the exterior of a given angle, draw another angle congruent to it. Can you do this in the interior of the given angle?*

No. $\S 14$ congruent angles states that the congruent angle to the given angle can be moved onto and identified with the given angle. Move the congruent angle onto the original angle such that one of their sides becomes identified and their interiors lie on the same side of the now identified sides. In accordance with congruence, their other sides will have become identified, and it is not possible for either side of the congruent angle to lie within the interior of the original angle while each angle has a side identified with the other.

Exercise 22: *How many common sides can two distinct angles have?*

2. A common side is a side from each of 2 distinct angles that have become identified with one another. An angle is defined as a figured formed by 2 rays eminating from the same point, known as sides. It follows that the maximum possible common sides is 2. On a given circle let there be an angle less than the full angle, and this angle has an exterior and interior region. Now let there be a second angle that has the same sides as the previously described angle, but its interior shall be the exterior of the previous. There are now 2 distinct angles with 2 common sides.

Exercise 23: *Can two non-congruent angles contain 55 angular degress each?*

If the question is understood as having two non-congruent angles that contain exactly 55 angular degrees each:

No. $\S 18$ angular degrees are all congruent. 2 angles that are both the sum of 55 congruent angular degrees are congruent to one another.

If the question is understood has having two non-congruent angles that contain at least 55 angular degrees each:

Yes. Following from the above, let there be $\angle \alpha$ that is congruent to the sum of 55 angular degrees. Now let there be $\angle \beta$ that is the sum of an angle congruent to $\angle \alpha$ and 1 angular degree. $\angle \alpha \neq \angle \beta$ but they both contain 55 angular degrees.

Exercise 24: *Can two non-congruent arcs contain 55 circular degrees each? What if these arcs have the same radius?*

If the question is understood as having two non-congruent arcs that contain **exactly** 55 circular degrees each:

Yes. $\S 18$ circular degrees are 1/360 part of a given circle and congruent to each other. $\S 10$ congruent arcs must come from the same or congruent cricles. Let there be a circle $A$ and a circle $B$ of different radii and both are divded respectively into 360 circular degrees. A sum of 55 circular degrees of circle $A$ cannot by moved onto and identified with the sum of 55 circular degrees of circle $B$.

No. $\S 18$ circular degrees of a given circle are all congruent. 2 arcs of the same circle that are both the sum of 55 congruent circular degrees are congruent to each other by $\S 10$.

If the question is understood as having two non-congruent arcs that contain **at least** 55 circular degrees each:

The second part of the preceding answer would change. Yes, two non-congruent arcs of the same radius containing 55 circular degrees can be non-congruent. On a given circle let there be $\stackrel{\frown}{a}$ that is the sum of 55 congruent circular degrees and $\stackrel{\frown}{b}$ that is the sum of an arc congruent to $\stackrel{\frown}{a}$ and 1 circular degree. By $\S 10$ there are now two non-congruent arcs that contain at least 55 circular degrees.

Exercise 25: *Two straight lines intersect at an angle containing 25 degrees. Find the measures of the remaining three angles formed by these lines.*

The three remaining angles measure 25, 155, and 155 degrees.

Let the straight lines be labeled $a$ and $b$ respectively and their point of intersection be $O$. Line $a$ has a point on either side of point $O$, $C$ and $D$. Line $b$ has a point on either side of point $O$, $M$ and $N$. $\angle{COM}$ is the given $25 ^{\circ}$ angle.

The intersection at $O$ of line $a$ creates 2 half lines eminating from $O$ in opposite directions that are continuations of each other on line $a$, $\overrightarrow{OC}$ and $\overrightarrow{OD}$. This is the straight angle $\S 16$. Thus $\angle{MOD}$ = the straight angle - $25 ^{\circ}$.

Applying this same technique on line $b$, which contains the 2 half lines $\overrightarrow{OM}$ and $\overrightarrow{ON}$, $\angle{DON}$ = the straight angle - $\angle{MOD}$ = the straight angle - (the straight angle - $25 ^{\circ}$) = $25 ^{\circ}$. Then $\angle{DON} = \angle{COM} = 25 ^{\circ}$.

Applying it yet again to line $b$, $\angle{CON}$ = the straight angle - $\angle{COM}$ = the straight angle - $25 ^{\circ}$. Then $\angle{COM} = \angle{MOD}$ = the straight angle - $25 ^{\circ}$.

By $\S 18$ there are 360 angular degrees about point $O$. $360 ^{\circ} - 25(\angle{DON}) - 25(\angle{COM}) = 310 ^{\circ}$. Since $\angle{COM} = \angle{MOD}$, $\angle{COM} + \angle{MOD}$ = $\angle{COM} + \angle{COM}$ = the remaining $310 ^{\circ}$. Thus $\angle{COM}$ and $\angle{MOD}$ are equal to half of the remaining $310 ^{\circ}$ or $155 ^{\circ}$ degrees each.

Exercise 26: *Three lines passing through the same point divide the plan into six angles. Two of them turn out to contain 25 degrees and 55 degrees respectively. Find the measures of the remaining four angles.*

This is similar work to exercise 25. The four remaining angles measure $100 ^{\circ}$, $25 ^{\circ}$, $55 ^{\circ}$, and $100 ^{\circ}$.

Exercise 27: *Using only a compass, construct a 1 degree arc on a circle, if a 19 degree arc of this circle is given.*

$\S 18$ the circle contains $360 ^{\circ}$. The product of 19 and the $19 ^{\circ}$ arc is a $361 ^{\circ}$ arc. This $361 ^{\circ}$ arc - $360 ^{\circ}$ full angle = $1 ^{\circ}$ arc.

Set the compass step to the distance of the chord subtending the given $19 ^{\circ}$ arc ($\stackrel{\frown}{AB}$). Starting at point $A$ on the circle and in the direction exterior to $\stackrel{\frown}{AB}$, mark a point on the circle, which will create a chord congruent to $\overline{AB}$. Repeat this along the circle, always placing the compass pin on the previously marked point, until a point is marked that creates a chord which intersects radius $\overrightarrow{OA}$. This final point, $C$, creates $\stackrel{\frown}{OC}$ which is a $1 ^{\circ}$ arc.