*Magic of Mathematics*:

"As nature puts forth its wonders in the garden, most people are oblivious to the massive calculations and mathematical work that have become so routine in nature."

Here Pappas wraps up the story about of the gardener who is surprised by all of the mathematics that appears in her garden. On this page she writes about a

*triple junction*-- a point where three line segments meet and the angles at the intersection are each 120 degrees. According to Pappas, triple junctions are found in soap bubbles, as well as the cracking of earth or stone and on kernels on a cob of corn (bringing this back to the garden).

Lesson 15-6 of the U of Chicago text is on "Angles Formed by Tangents." 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:

*O*, Line

*BC*tangent to Circle

*O*

Prove: Angle

*ABC*= Arc

*AB*/2

Proof:

Statements Reasons

1. Draw diameter

*B*,

*O*) there is exactly one line.

2. Arc

*AD*= 180 -

*AB*2. Arc Addition Postulate

3.

*CB*perpendicular

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

*. 1. Through any two points there is exactly one line.*~~AC~~

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

*P*),

*T*on Circle

*O*(opposite side of

*P*)

Prove: Angle

*P*= (Arc

*QTR*-

*QSR*)/2

Proof ("between two tangents"):

Statements Reasons

1. Draw

*B*,

*O*) 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.

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