Answer:
12. X = 100°, Y = 80°, Z = 160°
13. x = 65°
Step-by-step explanation:
Lets revise some facts in the circle
- The measure of the angle formed from the intersection of a tangent and a chord in a circle is equal to one-half the measure of the intercepted arc of the chord
- When two chords, intersect inside a circle then the measure of the angles formed is one-half the sum of the measures of the intercepted arcs.
Now lets solve the two problems
12.
∵ Y is an angle formed from the intersection of a chord and a tangent
∴ Y = [tex]\frac{1}{2}[/tex] the measure of the intercepted arc Z
∵ The measure of opposite arc to angle x is 200°
∵ The measure of the circle is 360°
∴ The measure of the arc Z = 360 - 200
∴ The measure of the arc Z = 160°
∴ Z = 160°
Substitute it in the expression of m∠Y
∴ Y = [tex]\frac{1}{2}[/tex] × 160°
∴ Y = 80°
∵ X and Y are formed from the intersection of line and ray
∴ X + Y = 180° ⇒ linear pair of angles
∵ Y = 80°
∴ X + 80 = 180
- Subtract 80 from both sides
∴ X = 100°
13.
∵ x is formed from the intersection of two chords in a circle
- That means the value of x is one-half the sum of the measures
of the two intercepted arcs
∴ x = [tex]\frac{1}{2}[/tex] the sum of the measures of the intercepted arcs.
∵ The measures of the intercepted arcs are 100° and 30°
∴ x = [tex]\frac{1}{2}[/tex] (100 + 30)
∴ x = [tex]\frac{1}{2}[/tex] (130)
∴ x = 65°
