The Theorem: An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc
This means that the measure of arc ABC (the purple portion of the circle itself) is twice the measure of angle C.
Note: Like inscribed angles, when the vertex is on the circle itself, the angle formed is half the measure of the intercepted arc.
Practice Problems
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 170°.
Therefore $$ x = \frac{1}{2} \cdot 170 = 85^{\circ} $$
Total measure of circle's circumference = 360°
$$ \frac 3 4 (360^{\circ}) = 270^{\circ} $$
By our theorem, we know that the angle formed by a tangent and a chord must equal half of the intercepted arc so $$x = \frac 1 2 \cdot 270^{\circ} =135^{\circ} $$
Circle 1 is the only circle whose intercepted arc is half the measure of the angle between the chord and the intersecting line.
The key to this problem is recognizing that $$ \overline{AOC}$$ is a diameter .
Therefore, $$m \overparen{ABC} = 180 ^{\circ} $$
At this point, you can use the formula, $$ \\ m \angle ACZ= \frac{1}{2} \cdot 180 ^{\circ} \\ m \angle ACZ= 90 ^{\circ} $$
The key to this problem is recognizing that the total degrees in a circle is $$ 360^{\circ} $$
From there you can set up an equation using the 3: 2 ratio.
$$ 3x + 2x =360 \\ 5x = 360 \\ \frac{5x}{5} = \frac{360}{5} \\ x = 72 \\ \overparen {JLM }= 2x = 2 \cdot 72 \\ \overparen {JLM }=144 ^{\circ} $$
At this point, you can use the formula, $$ \\ m \angle MJK= \frac{1}{2} \cdot 144 ^{\circ} \\ m \angle MJK = 72 ^{\circ} $$

Related Pages:
 Circles forumula, graph, equations
 Equation of A Circle
 Circumference
 Area
 Chord
 Tangent
 Arc of A Circle
 Intersecting Chords
 Inscribed Angle
 Secant of circle
 2 Tangents from 1 point
 Central Angle
 Angles, Arc, Secants, tangents
 Tangents, Secants and Side Lengths
 Tangent and a Chord
 images