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Friday, November 30, 2012

Proving Two Parallel Lines - Circles

Category: Plane Geometry

"Published in Newark, California, USA"

If two circles are tangent externally and a line is drawn through a point of contact and terminated by the circles. Prove that the radii drawn to its extremities are parallel.

Photo by Math Principles in Everyday Life
Solution:

Consider the given figure


Photo by Math Principles in Everyday Life

Proof:

1. Statement: ∠1 ≅ ∠2
    
    Reason: Vertical angles are congruent.

2. Statement: OP OA and O'P O'B
    
    Reason: All points in a circle are equidistant from its center.

3. Statement: AB is drawn through point P.
    
    Reason: Given item.

4. Statement: ΔOAP and ΔO'PB are isosceles triangles.

    Reason: An inscribed triangle in a circle which consist of a center of a circle and the two end points of a chord is always an isosceles triangle.

5. Statement: ∠1 ≅ ∠3 and ∠2 ≅ ∠4

    Reason: The two opposite angles of an isosceles triangle are congruent.

6. Statement: ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4

    Reason: Transitive property of congruence.

7. Statement: ∠AOP = 180º - (∠1 + ∠3)
                      ∠PO'B = 180º - (∠2 + ∠4)

    Reason: The sum of the interior angles of a triangle is 180º.

8. Statement: ∠AOP ≅ ∠PO'B

    Reason: By computation at #7, if ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4, then ∠AOP ≅ ∠PO'B.

9. Statement: OA ║ O'B

    Reason: If a transveral line (OO') passed the two alternating interior angles (∠AOP and ∠PO'B) that are congruent, then it follows that the two lines (OA and O'B) which are adjacent to the alternating interior angles are parallel.