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

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.

I know I haven't written about the Third Edition during the side-along reading, so let's catch up with that now. Since Chapter 13 of the old text has been split into different chapters in the new, the old Chapter 15 is the new Chapter 14. The first two lessons of the two chapters are identical, but Lesson 15-3 of the old text appears much earlier in the new -- Lesson 6-3. (This is a huge change, but that day I wrote only about Wickelgren's proof of the Inscribed Angle Theorem.) Lesson 15-4 of the old text has no real counterpart in the new text -- instead, Lesson 14-3 is the old Lesson 15-5. Finally, the new Lesson 14-4 is the old Lesson 13-5 (yes, one of those old Chapter 13 lessons). This move makes sense because now the two lessons on tangents to circles are in the same chapter.

Meanwhile, today in Economics class, I finally played John Stossel in its entirety to periods 1-3. I might as well play it today, because today's lesson plan is also impossible to fulfill -- I'm supposed to play the 1990's film Wall Street on DVD today, but the disc doesn't work in fifth period (the one Econ class that already finished the Stossel assignment). With nothing else to do, the students immediately begin a game of -- Heads Up Seven Up??? (Yes, these are high school seniors.)

By the way, right after playing Stossel's Greed, YouTube autoplayed another Stossel video -- and this one is directly related to education. In this video, the journalist argues that public schools don't teach the basics properly. Though his main point is that charter schools are better at this than district schools, near the end of the first part, he also mentions tutoring centers such as Sylvan. Traditionalists would agree that tutoring centers teach the basics (though here Stossel is referring to reading rather than arithmetic). But our traditionalists wouldn't necessarily agree that charters teach better -- after all, the charter I taught at last year used Illinois State Project-Based Learning. This is all I'll say about charters and traditionalists, since I just had a post on traditionalists yesterday.

This is what I wrote last year 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.