Finally, the last relationship you will explore is found when tangents and secants of a circle intersect outside of the circle.  Once again, using your knowledge of intersecting lines both inside and outside of a circle, write out a hypothesis for the relationship between the angle formed by the intersection of a tangent and a secant line and the measures of the intersect arcs. Write this hypothesis in your notes.

Using the Intersecting Secant and Tangent applet linked below, click and drag the endpoints of AC  so that you always have the red and blue arcs in opposite places on the circle. To change the size of your circle, click and drag point A or point E. Observe the changes in m∠D as you change the arc measurements.

Click on the image below to open the applet. Manipulate the angle so that m AC  is 110 degrees, then 130 degrees, and finally 170 degrees.

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Copy the table below into your notes. Then use the measures of the arcs to complete the table, and answer the questions below.

m ∠ D	m AC	m AB	m AC – m AB
	110°
	130°
	170°

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Check Your Answer

• How does m∠S compare to m AC – m AB?

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  Check Your Answer

  m ∠D = 1 over 2 1 2 (m AC – m AB)

• If ∠D is the angle created between an intersecting tangent and secant, and AC  and AB  are the intercepted arcs of the tangent and secant, what can you say about the measure of ∠D and the measures of AC  and AB?

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  Check Your Answer

  If a tangent and a secant intersect at a common point, then the measure of the angle formed is equal to half the difference of the measures of the intercepted arcs.

Pause and Reflect

How does the relationship that you discovered for a tangent and a secant that intersect outside a circle compare to the relationships you discovered for two intersecting tangents and two secants that intersect outside a circle?

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Check Your Answer

The relationships are very similar in that the measure of the angle of intersection is equal to one-half the difference of the measures of the intercepted arcs.

Practice

1. In circle K below, what is m∠DAC?
   

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   The measure of the angle between an intersecting tangent and secant is one-half the difference of the measures of the intercepted arcs.

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   Check Your Answer

   m∠DAC = 1 over 2 1 2 (m CD – m BD)
   m∠DAC = 1 over 2 1 2 (174.4° – 83.5°)
   m∠DAC = 45.4°

2. In circle Q below, what is m ST?
   

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   The measure of the angle between an intersecting tangent and secant is one-half the difference of the measures of the intercepted arcs.

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   Check Your Answer

   m∠SRT = 1 over 2 1 2 (m PT – m ST)
   25° = 1 over 2 1 2 (101.7° – m ST)
   25° = 50.85° – 1 over 2 1 2 m ST
   1 over 2 1 2 m ST = 50.85° – 25°
   m ST = 51.7°

3. A cylindrical building is viewed from above as shown in the diagram. An observer standing at point R knows that if she looks directly at the center of the building, along RP, that m PT along the backside of the building will be 120°. What is the angle between the observer’s line of sight toward the center of the building and a tangent line toward the edge of the building?
   

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   The measure of the angle between an intersecting tangent and secant is one-half the difference of the measures of the intercepted arcs.

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   Check Your Answer

   m∠SRT = 1 over 2 1 2 (m PT – m ST)
   First you need to determine, m ST.
   m PT + m ST = 180°
   120° + m ST = 180°
   m ST = 60°
   m∠SRT = 1 over 2 1 2 (m PT – m ST)
   x° = 1 over 2 1 2 (120° – 60°)
   x = 30