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Bellwork  A 10 foot piece of wire is cut into two pieces. One piece is bent to form a square. The other forms a circle inscribed in the square. How long.

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Presentation on theme: "Bellwork  A 10 foot piece of wire is cut into two pieces. One piece is bent to form a square. The other forms a circle inscribed in the square. How long."— Presentation transcript:

1 Bellwork  A 10 foot piece of wire is cut into two pieces. One piece is bent to form a square. The other forms a circle inscribed in the square. How long is each piece?  Solve for x 5x 2 -13x-6=0 Clickers

2 Bellwork Solution  A 10 foot piece of wire is cut into two pieces. One piece is bent to form a square. The other forms a circle inscribed in the square. How long is each piece?  Solve for x 5x 2 -13x-6=0

3 Section 10.4

4 The Concept  Yesterday we discussed how chords can create arcs  Today we’re going to discuss the concept of inscribed angles and polygons  We’ve dealt with inscribing somewhat (chapter 11), but today we’re going to completely define how they operate

5 Definition  Inscribe To draw within an object  Circumscribe Draw around an object  Inscribed Angle Angle whose vertex is on the circle and sides are chords of a circle  Intercepted Arc Arc formed by an inscribed angle Inscribed Angle Intercepted Arc

6 Theorem Theorem 10.7 The measure of an inscribed angle is one half the measure of its intercepted arc Where does this come from. Let’s look at a the simple case: B A Cx

7 Example  Find the measure of arc AB  Find the measure of arc CD 55 o A C B 31 o D

8 On your own  What is the measure of angle ABC 82 o A C B D

9 On your own  What is the measure of arc CB 33 o A C B

10 On your own  What is the measure of Arc AC 62 o A C B D

11 Theorems Theorem 10.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent

12 On your own  Name two pairs of congruent angles in the figure JK M L

13 Theorems Theorem 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a right triangle and the angle opposite the diameter is the right angle. Why?

14 Theorems Theorem A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary

15 Example  What is the measure of angle D?  Angle A? A 95 o B C D 45 o

16 On your own  What is the measure of angle J? JK M L 81 o

17 On your own  Solve for x JK M L 2x+4 o 4x-1 o

18 On your own  A Parallelogram is inscribed in a circle. What is the measure of one of it’s angles?

19 Homework  , 2-24 even, 27-29

20 Most Important Points  Inscribe/Circumscribe Dichotomy  Inscribed Angles & Intercepted Arcs  Inscribed Right Triangles  Inscribed Polygons


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