Inscribed Angles and Quadrilaterals Unit 1: Circles and Volume

Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle INTERCEPTED ARC INSCRIBED ANGLE

Determine whether each angle is an inscribed angle Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 1. YES; CL C L O T

Determine whether each angle is an inscribed angle Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. NO; QVR 2. Q V K R S

To find the measure of an inscribed angle… 1600 800

What do we call this type of angle? What is the value of x? How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!! 120 x y

40  112  Examples 3. If m JK = 80, find m <JMK. 4. If m <MKS = 56, find m MS. 112  M Q K S J

If two inscribed angles intercept the same arc, then they are congruent. 72

x = 3 In J, m<3 = 5x and m<4 = 2x + 9. Example 5 Find the value of x. 3 Q D J T U 4 x = 3

If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

A circle can be circumscribed around a quadrilateral if and only if its opposite angles are supplementary. B A D C

y = 70 z = 95 110 + y =180 z + 85 = 180 Example 8 Find y and z. z 110

If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. 180 diameter

Example 6 In K, m<GNH = 4x – 14. Find the value of x. 4x – 14 = 90 H K x = 26 N G

x = 11 6x – 5 + 3x – 4 + 90 = 180 or 6x – 5 + 3x – 4 = 90 H K N G Example 7 In K, m<1 = 6x – 5 and m<2 = 3x – 4. Find the value of x. 6x – 5 + 3x – 4 + 90 = 180 or 6x – 5 + 3x – 4 = 90 H 2 K 1 N G x = 11