12-3 Inscribed Angles

Quick Review

An inscribed angle has its vertex on a circle and its sides are chords. An intercepted arc has its endpoints on the sides of an inscribed angle, and its other points in the interior of the angle. The measure of an inscribed angle is half the measure of its intercepted arc.

Example

What is m P S ⌢ ? m , modified p s with frown above . question mark  What is m ∠ R ? m angle r question mark

The m ∠ Q = 60 m angle , q equals 60  is half of m P S ⌢ , m , modified p s with frown above , comma  so m P S ⌢ = 120 . ∠ R m , modified p s with frown above . equals 120 . , angle r  intercepts the same arc as ∠ Q , angle q comma  so m ∠ R = 60 . m angle . r equals 60 .

Exercises

Find the value of each variable. Line l is a tangent.

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12-4 Angle Measures and Segment Lengths

Quick Review

A secant is a line that intersects a circle at two points. The following relationships are true:

a · b = c · d eh middle dot , b equals . c middle dot d

m ∠ 1 = 1 2 ( x + y ) m angle , 1 equals , 1 half . open x plus y close

( w + x ) w = ( y + z ) y open w plus . x close w equals open . y plus z close y

m ∠ B = 1 2 ( a − b ) m angle b equals , 1 half . open eh minus b close

( y + z ) y = t 2 open y plus . z close y equals . t squared

m ∠ B = 1 2 ( a − b ) m angle b equals . 1 half . open eh minus b close

Example

What is the value of x?

( x + 10 ) 10 = ( 19 + 9 ) 10 x + 100 = 252 x = 15.2 table with 3 rows and 2 columns , row1 column 1 , open x plus 10 close 10 , column 2 equals open 19 plus 9 close , row2 column 1 , 10 x plus 100 , column 2 equals 252 , row3 column 1 , x , column 2 equals , 15.2 , end table

Exercises

Find the value of each variable.

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