1 / 35

Geometry

Geometry. Inscribed Angles. Goals. Know what an inscribed angle is. Find the measure of an inscribed angle. Solve problems using inscribed angle theorems. ABC is an inscribed angle. AC is the intercepted arc. Inscribed Angle.

jackson-bolton
Download Presentation

Geometry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geometry Inscribed Angles

  2. Goals • Know what an inscribed angle is. • Find the measure of an inscribed angle. • Solve problems using inscribed angle theorems.

  3. ABC is an inscribed angle. AC is the intercepted arc. Inscribed Angle The vertex is on the circle and the sides contain chords of the circle. A B C

  4. Inscribed Angle How does mABC compare to mAC? A B C

  5. A R B O Draw circle O, and points A & B on the circle. Draw diameter BR.

  6. A R B O Draw radius OA and chord AR. 2 3 1

  7. 2 1 3 (Very old) Review • The Exterior Angle Theorem (4.2) • The measure of an exterior angle of a triangle is equal to the sum of the two remote, interior angles. m1 + m2 = m3

  8. A R B O mARO + mOAR = mAOB What type of triangle is OAR? 2 Isosceles 3 The base angles of an isosceles triangle are congruent. 1 1  2

  9. A R B O mARO + mOAR = mAOB • m1 + m2 = m3 • But m1 = m2 • m1 + m1 = m3 • 2m1 = m3 • m1 = (½)m3 2 1 3 This angle is half the measure of this angle.

  10. Where we are now. Recall: the measure of a central angle is equal to the measure of the intercepted arc. A 2 x (x/2) x 3 1 R B O m1 = (½)m3

  11. Theorem 12.8 If an angle is inscribed in a circle, then its measure is one-half the measure of the intercepted arc. A x (x/2) R B O Inscribed Angle Demo

  12. Example 1 44 ? 88

  13. Example 2 A ? 170 B 85 C

  14. Example 3 ? 60 The circle contains 360. 360 – (100 + 200) = 60 100 30 x 200

  15. Another Theorem 2x ? Theorem 10.9 If two inscribed angles intercept the same (or congruent) arcs, then the angles are congruent. x ? x Theorem Demonstration

  16. A very useful theorem. Draw a circle. Draw a diameter. Draw an inscribed angle, with the sides intersecting the endpoints of the diameter.

  17. A very useful theorem. What is the measure of each semicircle? 180 What is the measure of the inscribed angle? 90 90

  18. Theorem 12.10 If an angle is inscribed in a semicircle, then it is a right angle. Theorem 12.10 Demo

  19. Theorem 12.2: Tangent-Chord If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of the intercepted arc. B C 2 1 A

  20. a b 1 2 Simplified Formula

  21. Example 1 B C 160 200 80 A

  22. Example 2. Solve for x. B C (10x – 60) 4x A

  23. Inscribed Polygon • The vertices are all on the same circle. • The polygon is inside the circle; it is inscribed.

  24. Cyclic Quadrilaterals

  25. A A cyclic quadrilateral has all of its vertices on the circle. B D C

  26. B C A An interesting theorem. D

  27. B C A An interesting theorem. D

  28. B C A An interesting theorem. Adding the equations together… D

  29. B C A An interesting theorem. D

  30. An interesting theorem.

  31. B C A An interesting theorem. D BAD and BCD are supplementary.

  32. 1 2 4 3 Theorem 12.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Theorem 10.11 Demo m1 + m3 = 180 & m2 + m4 = 180

  33. Example Solve for x and y. 4x + 2x = 180 6x = 180 x= 30 and 5y + 100 = 180 5y = 80 y = 16 2x 5y 4x 100

  34. Summary • The measure of an inscribed angle is one-half the measure of the intercepted arc. • If two angles intercept the same arc, then the angles are congruent. • The opposite angles of an inscribed quadrilateral are supplementary.

  35. Inscribed Hexagon Practice Problems

More Related