Question

Find the value of z, secant and tangent angles

Answer 1

`A)\ \ z = 110`

Explanation:

One is given a circle with two secants. Please note that a secant is a line that intersects a circle in two places. The secant exterior angle theorem states that when two secants intersect each other outside of a circle, the measure of the angle formed is half the of the positive difference of the intersecting arcs. One can apply this theorem here by stating the following:

`2x+15=((10x+20)-(80))/(2)`

Simplify,

`2x+15=((10x+20)-(80))/(2)`

`2x+15=(10x-60)/(2)`

`2x+15=5x-30`

Inverse operations,

`2x+15=5x-30`

`15=3x-30`

`45=3x`

`15=x`

Substitute this value into the equation for one of the intersecting arcs to find the numerical value of that intersecting arc:

`10x+20\\x = 15\\\\10(15) + 20\\= 150 + 20\\= 170`

The sum of all arc measures in a circle is (360) degrees. One can apply this here by stating the following:

`80 + 170 + z = 360`

Simplify,

`80 + 170 + z = 360`

`250 + z = 360`

Inverse operations,

`250 + z = 360`

`z = 110`

Answer 2

z = 110

Explanation:

The measure of an angle created by the intersection of two secants outside a circle is half the difference of the angles it intercepts. In this it would be:

2x + 15 = 1/2 * (10x + 20 - 80)

We can now solve:

4x + 30 = 10x - 60

30 = 6x - 60

6x = 90

x = 15

This means the values of 10x + 20 is 10(15) + 20, which is 170.

Now, we can add up all the arcs in a circle, which sum to 360 degrees:

360 = z + 170 + 80

360 = z + 250

z = 110