Question

HELP GEOMETRY find x,y,z

Answer 1

We have three problems that are basically the same, so let's try to come up with a general formula.

As you've observed in the previous problems I've helped you with, the angle cut off by a tangent and a chord is half that of the arc they cut off. So, in the first two problems, if we let the angle we're given be
`a`, we have
`x = 2a`.

Then, by the straight-line angle rule (angles that together form a straight line sum to 180 degrees),
`z = 180 - a`. Applying the first rule again,
`y = 2z = 360 - 2a`.

Applying these formulas, for the first question,
`x = 115 \cdot 2 = 230`,
`y = 180 - x = 65`, and
`z = 360 - 2a = 130`.

For the second problem,
`x = 180`,
`z = 90`, and
`y = 180`.

For the third, the variables are positioned differently, so we replace
`x` with
`z` (because
`z` is now the arc cut off by our given angle) and
`z` with
`x`.

So,
`z = 2 \cdot 134 = 268`,
`x = 180 - 134 = 46`, and
`y = 2x = 92`.