Question

Help anyone I’m really struggling in this

Answer 1

12. X = 100°, Y = 80°, Z = 160°

13. x = 65°

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 =
`(1)/(2)` 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 =
`(1)/(2)` × 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 =
`(1)/(2)` the sum of the measures of the intercepted arcs.

∵ The measures of the intercepted arcs are 100° and 30°

∴ x =
`(1)/(2)` (100 + 30)

∴ x =
`(1)/(2)` (130)

∴ x = 65°