Arcs and Angles

1. Arcs and Angles Class Investigation

2. Geometry 1/11/11 • Get out HW, pencil, binder, and begin Do Now! • Objectives: • Describe and apply conjectures of inscribed angles AGENDA • Do Now! (5 min) • Inscribed angles (25 min) • Practice problems (20)

3. Vocabulary • Inscribed Angle: An angle created by 2 chords (<A) where the vertex lies on the circle • Semicircle: Half a circle • Secant: A line the intersects the circle at 2 points A Secant Line

4. Inscribed angle review We know: Inscribed angle = ½ (arc) C A 60º B

5. Inscribed angles • An angle inscribed in a semicircle is a right angle. C 90º A B

6. Investigation 1 • Step 1: Pick two points on the edge of your circle and label them A and B • Step 2: Now pick a point that is not between A and B and label it point P • Step 3: Now draw the chords AP and BP and measure inscribed angle <APB • <APB = ____________ • Step 4: Now draw another point that is not between A and B and label it point Q • Step 5: Draw the chords AQ and BQ, and measure inscribed angle <AQP • <AQB = ____________ • Step 7: What do you notice about how <APB and <AQB compare?? A B P Q

7. Inscribed Angles Intercepting Arcs Conjecture • Inscribed angles that intercept the same arc are congruent.

8. Investigation 2 • Step 1: Now draw 4 points on the edge of your circle and label them A, B, C and D • Step 2: Now connect all four points with lines AB, BC, CD, and DA • Step 3: Now measure angles <A, <B, <C, and <D and add them all up • <A = _______ • <B = _______ • <C = _______ • <D = _______ D A C B

9. Cyclic Quadrilateral Conjecture • A quadrilateral inscribed in a circle is called a cyclic quadrilateral • The opposite angles of a cyclic quadrilateral are supplementary.

10. Parallel Lines Intercepted Arcs Conjecture • Parallel lines intercept congruent arcs on a circle. 43° 43°