Note: Like inscribed angles, when the vertex is on the circle itself, the angle formed is half the measure of the intercepted arc.
Chord $$ AC $$ intercepts a tangent tangent at point C. If the measure of $$ \overparen = 70^ $$, what is x ? Show Answer
Use the theorem above to find the measure of angle formed by the intersection of the tangent that intersects chord AC . By the theorem, the measure of angle is half of the intercepted arc which is $$70 ^$$ .
Chord $$ AC $$ intercepts a tangent tangent at point C. If the measure of $$ \overparen = 110^ $$, what is x ? What is the $$ m\overparen $$(the measure of arc ABC) ? Answer
Remember the theorem: the angle formed by the tangent and the chord is half of the measure of the intercepted arc. Therefore, the arc is double the angle. $$ m\overparen = 2 \cdot 110^=55^ $$
For the $$ m\overparen
Total measure of circle's circumference = 360°. $$ \frac 3 4 (360^) = 270^ $$ By our theorem, we know that the angle formed by a tangent and a chord must equal half of the intercepted arc so $$x = \frac 1 2 \cdot 270^ =135^ $$ .
Look at Circle 1 and Circle 2 below. In only one of the two circles does a tangent intersect with a chord. Which circle is it? Answer
Circle 1 is the only circle whose intercepted arc is half the measure of the angle between the chord and the intersecting line.
What is the measure of $$ \angle ACZ $$ for circle with center at O? Answer
The key to this problem is recognizing that $$ \overline
$$ \overparen
The key to this problem is recognizing that the total degrees in a circle is $$ 360^ $$ . From there you can set up an equation using the 3:2 ratio. $$ 3x + 2x =360 \\ 5x = 360 \\ \frac = \frac \\ x = 72 \\ \overparen = 2x = 2 \cdot 72 \\ \overparen =144 ^ $$ At this point, you can use the formula, $$ \\ m \angle MJK= \frac \cdot 144 ^ \\ m \angle MJK = 72 ^ $$
