Lesson 15-6: Angles Formed by Tangents (Day 156)

Today I subbed in a high school self-contained special ed class. It's in my new district. This is obviously an unrepresentative class, and so there's no need to do "A Day in the Life" today.

While several of the students are working on math, it's mostly on a very familiar website, IXL. None of them seem to need my help with their math IXL assignments.

Most of the day is spend preparing for an upcoming field trip to Catalina Island. For non-Californian blog readers, I refer here to the 1950's song "26 Miles (Santa Catalina)" by the Four Preps. Yes, Catalina is an island that's approximately that distance from the mainland. I've never been there, but apparently, these students are going there soon. They will take a ferry that takes about an hour to traverse that distance. One guy, an Honors Biology student with an interest in marine life, is especially looking forward to the trip.

Another guy definitely has issues, though. (As usual, I don't disclose these issues on the blog.) So a pair of aides and I take him out for a walk. On this walk, I sing "Mousetrap Car Song" again -- I didn't plan on this becoming my new walking song now that the Big March is over, but the guy seems to enjoy it and tries to sing part of it with me.

Today is Fiveday on the Eleven Calendar:

Resolution #5: We treat people who are great at math as heroes.

Of course, I don't have much opportunity to fulfill this resolution today. The original version of this resolution referred to the year 1955. Well, the Catalina song I mentioned earlier is from the 1950's, albeit a few years after 1955.

Today is Thursday, and so it's time for my weekly series on COVID-97 and high school Track. This week, my alma mater participated in the third dual meet of the season -- the second league meet. It was held on our home track.

But as I mentioned in last Friday's post, the league combines the two smallest schools -- and those two were our opponents last week. As expected, neither of those school has very many participants -- indeed not in the distance races. (They might have had some sprinters, though.)

And so our school runs the 1600 races unopposed. There is only one girls and one boys 1600 race (as opposed to having separate Varsity and Junior Varsity races). The top boys appear to run just under five minutes, with some runners closer to my own best times.

With fewer races, the meet progresses quickly, and so I decided to stay through the 800's. The top boys in this race are just over two minutes. I don't remember my own 800 times as well as my 1600 times, but I believe that my time was closer to the mid-2:00 range. This would have placed me much closer to the middle of the pack.

On April 29th, 1999 (third dual meet, second league meet), my 1600 time would have been 5:12.

This would have put me in fourth place in the combined 1600 race.

Lesson 15-6 of the U of Chicago text is called "Angles Formed by Tangents." In the modern Third Edition of the text, angles formed by tangents appear in Lesson 14-5.

This is what I wrote two years ago about today's lesson:

The theorems in this lesson are similar to those in yesterday's lesson.

Tangent-Chord Theorem:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Given: AB chord of Circle O, Line BC tangent to Circle O
Prove: Angle ABC = Arc AB/2

Proof:
Statements                                     Reasons
1. Draw diameter BD.                    1. Through any two points there is exactly one line.
2. Arc AD = 180 - AB                    2. Arc Addition Postulate
3. CB perpendicular BD                 3. Radius-Tangent Theorem
4. Angle ABC = 90 - ABD             4. Angle Addition Postulate
5. Angle ABC = 180/2 - Arc AD/2 5. Inscribed Angle Theorem
6. Angle ABC = (180 - Arc AD)/2 6. Distributive Property
7. Angle ABC = Arc AB/2              7. Substitution Property of Equality

Tangent-Secant Theorem:
The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs.

Given: Line AB secant, Ray EC tangent at point C, forming Angle E,
Arc AC = x, Arc BC = y
Prove: Angle E = (x - y)/2

Proof ("between a tangent and a secant"):
Statements                                     Reasons
1. Draw AC.                                   1. Through any two points there is exactly one line.
2. Angle DCA = x/2, EAC = y/2     2. Inscribed Angle Theorem
3. Angle DCA = EAC + E              3. Exterior Angle Theorem
4. Angle E = DCA - EAC               4. Subtraction Property of Equality
5. Angle E = x/2 - y/2                     5. Substitution Property of Equality
6. Angle E = (x - y)/2                     6. Distributive Property

In the text, the "between two tangents" is given as an exercise. The Given part of this proof with the way the points are labeled is completely different from the first part.

Given: Ray PV tangent at Q, Ray PU tangent at R
S on Circle O (same side of QR as P), T on Circle O (opposite side of QR as P)
Prove: Angle P = (Arc QTR - QSR)/2

Proof ("between two tangents"):
Statements                                     Reasons
1. Draw QR.                                   1. Through any two points there is exactly one line.
2. Angle VQR = Arc QTR/2,          2. Inscribed Angle Theorem
    Angle PQR = Arc QSR/2
3. Angle VQR = PQR + P              3. Exterior Angle Theorem
4. Angle P = VQR - PQR               4. Subtraction Property of Equality
5. Angle P = Arc QTR/2 - QSR/2  5. Substitution Property of Equality
6. Angle P = (Arc QTR - QSR)/2   6. Distributive Property

In some ways, the Tangent-Chord Theorem is just like yesterday's Angle-Chord Theorem, except that one of the intercepted arcs is 0 degrees. The bonus question concerns a solar eclipse.