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 transcript:

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

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

Section 10.4

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

Homework  , 2-24 even, 27-29

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