Tangent of a Circle Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. AB OP O B P A If the tangent meets a radius, then 

Converse of Tangent of a Circle Theorem If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. AB is tangent to O. A P B O If  then tangent

Tangent Segments to a Circle The two segments tangent to a circle from a point outside the circle are congruent. AB CB B A O C

Theorem (from 10.1-2) In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs. Theorem (from 10.1-2) In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord. Theorem (from 10.1-2) In a circle, the perpendicular bisector of a chord contains the center of the circle.

mB = ½ mAC Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. mB = ½ mAC A B C

Which of these is an inscribed angle? NOT Inscribed Inscribed Inscribed NOT Inscribed

Corollaries to the Inscribed Angle Theorem You will use three corollaries to the Inscribed Angle Theorem to find the measures of angles in circles. Corollaries to the Inscribed Angle Theorem Two inscribed angles that intercept the same arc are congruent. 2. An angle inscribed in a semicircle is a right angle. 3. The opposite angles of a quadrilateral inscribed in a circle are supplementary. 1 2 3 4

Which pairs of angles are congruent? 3 1 2 4

What is the measure of the numbered angles? 1 2

What is the relationship between angles 1 & 2, and angles 3 & 4? They are supplementary

mC = ½ mBDC B B D D C C Tangent Angles Theorem The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. mC = ½ mBDC B B D D C C

Name the intercepted arc F B A G E D H C S X Q V T R J K N L M

Inside Angles Theorem: The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. m1 = ½ (x + y) Outside Angles Theorem: The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. m  1 = ½ (x – y) x° y° 1 1 1 y° y° 1 y° x° x° x°

Theorem (10.1-2 Equations) For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle. I. II. III. a t c x w b y y d z z a • b = c • d (w + x)w = (y + z)y (y + z)y = t2