Question

Please help!!

I don't understand this can you please explain how you found the answer as well thanks​

Answer 1

x = 140°

Explanation:

Circle Theorem vocabulary

• Secant: a straight line that intersects a circle at two points.

• Arc: the curve between two points on the circumference of a circle.

• Intercepted arc: the curve between the two points where two chords or line segments (that meet at one point on the other side of the circle) intercept the circumference of a circle.

Intersecting Secants Theorem

If two secant segments are drawn to the circle from one exterior point, the measure of the angle formed by the two lines is half of the (positive) difference of the measures of the intercepted arcs.

The secant segments on the given diagram are the blue lines.

The measure of the angle formed by the two lines at the exterior point is 10°. According to the Intersecting Secants Theorem, this is half of the difference of the measures of the intercepted arcs labelled 30° and y (see attachment).

Therefore:

`\implies (1)/(2)(30^(\circ)-y)=10^(\circ)`

`\implies 30^(\circ)-y=20^(\circ)`

`\implies y=10^(\circ)`

An arc measure equals its corresponding central angle measure. Angles around a point sum to 360°. As one of the secant segments passes through the diameter of the circle, the sum of arcs 30°, y and x is 180°.

Therefore:

`\implies 30^(\circ)+y + x = 180^(\circ)`

`\implies 30^(\circ)+10^(\circ) + x = 180^(\circ)`

`\implies 40^(\circ) + x = 180^(\circ)`

`\implies x = 140^(\circ)`