Question

You know the measure of the exterior angle which forms a linear pair with the vertex angle. Describe two ways you can find measures of the interior angles of the triangle.

Answer 1

Explanation:

A linear pair are two given angles whose sum is equal to
`180^(o)`.

Let the exterior angle be represented by
`x^(o)`, and the three interior angles of the triangle be represented by
`a^(o)`,
`b^(o)` and
`c^(o)`.

Assume that the vertex angle
`c^(o)` is the linear pair to the exterior angle
`x^(o)`.

i.e
`x^(o)` +
`c^(o)` =
`180^(o)` (sum of angles on a straight line)

Thus,

i.
`a^(o)` +
`b^(o)` =
`x^(o)` (sum of two opposite interior angles is equal to an exterior angle)

⇒
`a^(o)` =
`x^(o)` -
`b^(o)`

Also,

`b^(o)` =
`x^(o)` -
`a^(o)`

But,

`a^(o)` +
`b^(o)` +
`c^(o)` =
`180^(o)` (sum of angles in a triangle)

⇒
`c^(o)` =
`180^(o)` - (
`a^(o)` +
`b^(o)`)

and

`a^(o)` +
`b^(o)` =
`180^(o)` -
`c^(o)`

ii.
`a^(o)` +
`b^(o)` +
`c^(o)` =
`180^(o)` (sum of angles in a triangle)

`a^(o)` =
`180^(o)` - (
`b^(o)` +
`c^(o)`)

Also,

`a^(o)` +
`b^(o)` =
`x^(o)`

⇒
`b^(o)` =
`x^(o)` -
`a^(o)`

`c^(o)` =
`180^(o)` -
`x^(o)`