Non-Central Angles KEY

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Abigayle Woods
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1 ¼ mabc Non-Central Angles KEY Use the circle below and follow the directions on the next page. 2012, TESCCC 09/27/12 page 1 of 11

2 Non-Central Angles KEY Part I: Inscribed Angles 1. Fold the circle in half and draw the segment along the crease so that its endpoints are on the circle. Label the segment, AB. AB is called a chord for the circle. A chord is a segment whose AB endpoints are on a circle. Since divides the circle into two congruent pieces, it passes through the circle s center and is a type of chord called a diameter. Find the center of chord AB (the center of the circle) by folding. Then label the center point P. APB a. is called a central angle for the circle. A central angle is an angle whose vertex is m APB the center of a circle. What is the? m APB = 180. ACB b. Randomly locate point C on either semicircle and use a straightedge to draw. ACB The angle you drew,, is called an inscribed angle because its vertex lies on the m ACB circle. Use a protractor to find. Record your answer below. 0 m ACB = 90. c. How does the measure of the inscribed angle compare to that of the central angle? The measure of the inscribed angle is half of that of the central angle. DE 2. Fold your paper so that point A and point B are directly on top of each other. Draw chord along the crease of your paper. m APE a. What is the? How do you know? 0 m APE = 90. The paper-folding divides the circle into quarters; therefore, the angle measures 90. (Students may also measure with a protractor.) ¼ABE b. Randomly select a point F that is on the major arc,. Use your straightedge to draw AFE m AFE the inscribed angle,. Use your protractor to find the. Record your answer below. 0 m AFE = 45 c. How does the measure of the inscribed angle compare to that of the central angle? The measure of the inscribed angle is half of that of the central angle. 3. Use your protractor to draw another central angle. a. What is the measure of the central angle that you drew? Record your answer below. Student answers will vary 2012, TESCCC 09/27/12 page 2 of 11

3 b. Use your straightedge to draw the inscribed angle for the central angle in part a. Use your protractor to find the measure of the inscribed angle. Record your answer below. Student answers will vary c. How does the measure of the inscribed angle compare to that of the central angle? The measure of the inscribed angle is half of that of the central angle. 4. Based on your responses to questions 1 through 3, write a conjecture about the measure of an inscribed angle. The measure of an inscribed angle is half of that of its central angle. 2012, TESCCC 09/27/12 page 3 of 11

4 Non-Central Angles KEY Part II: Angles Exterior to the Circle Use Figure 1 below and follow the directions on the next page. Figure 1 1 A B C D suur AC»AB BD suur and are secant lines for the circle because they intersect the circle in two points.»cd 1 and are intercepted arcs for the exterior angle,. 2012, TESCCC 09/27/12 page 4 of 11

5 Non-Central Angles KEY 5. Locate the center of the circle with paper folding. 6. Draw the central angles for»ab and angles for the corresponding arcs. Express the»cd. Use a protractor to find the measure of the central» mab and respective central angles. Record your answers in the table below.» mcd as the measure of their 7. Find the absolute difference of the measures of the arcs. Record your work in the table below. 8. Use a protractor to find the. Record your answer in the table below. 9. How does the compare to the absolute difference? is half the difference of the intercepted arcs mab» mcd» mab» mcd» , TESCCC 09/27/12 page 5 of 11

6 Non-Central Angles KEY Angles Exterior to the Circle: Use Figure 2 below and follow the directions on the next page. suur AC is a secant line for the circle because it intersects the circle in two points. BD suur is a tangent line for the circle because it intersects the circle in exactly one point.»ab»bc and are intercepted arcs for the exterior angle, 1. Figure 2 A B C D , TESCCC 09/27/12 page 6 of 11

7 Non-Central Angles KEY 10. Locate the center of the circle with paper folding. 11. Draw the central angles for»ab and mab» for the corresponding arcs. Express the and angles. Record your answers in the table below.»bc. Use a protractor to find the measure of the central angles» mbc as the measure of their respective central 12. Find the absolute difference of the measures of the arcs. Record your work in the table below. 13. Use a protractor to find the. Record your answer in the table below. 14. How does the compare to the absolute difference? is half the difference of the intercepted arcs mab» mbc» mab» mbc» , TESCCC 09/27/12 page 7 of 11

8 Non-Central Angles KEY Angles Exterior to the Circle: Use Figure 3 below and follow the directions on the next page. suur AD and one point. ¼ABC and Figure 3 A C B D 1 46 CD suur are tangent lines for the circle because each of the lines intersect the circle at exactly»ac are intercepted arcs for the exterior angle, , TESCCC 09/27/12 page 8 of 11

9 134 = , TESCCC 09/27/12 page 9 of 11

10 2012, TESCCC 09/27/12 page 10 of 11

11 15. Locate the center of the circle with paper folding. Non-Central Angles KEY»AC»AC 16. Draw the central angles for. Use a protractor to find the measure of the central angle for. mac» Express the as the measure of its central angle. Record your answer in the table below. The ¼ mabc, the major arc can be found by subtracting from 360. Express the degrees. Record your answer in the table below. ¼ mabc in terms of 17. Find the absolute difference of the measures of the arcs. Record your work in the table below. 18. Use a protractor to find the. Record your answer in the table below. 19. How does the compare to the absolute difference? is half the difference of the intercepted arcs mac» mac» mabc ¼ Based on your findings, write a conjecture about the exterior angle formed when two secant lines, a secant line and a tangent line, or two tangents intersect in the exterior of a circle. If the segments (two secants, a secant and a tangent, or two tangents) intersect in the exterior of a circle, the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. 2012, TESCCC 09/27/12 page 11 of 11

Name Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion

Name Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion Section. Lines That Intersect Circles Lines and Segments That Intersect Circles A chord is a segment whose endpoints lie on a circle. A secant is a line that intersects a circle at two points. A tangent