Question

The measure of one interior angle of a parallelogram is 30 degrees less than 9 times the measure of another angle. What is the measure of each angle?

Answer 1

The interior angles of the parallelogram are either:
`color(white)("XXX"){159^@,21^@,159^@,21^@}`
or
`color(white)("XXX"){3 3/4color(white)("")^@,176 1/4color(white)("")^@, 3 1/4color(white)("")^@, 176 1/4color(white)("")^@}`

Explanation:

Case 1: The given relation applies to consecutive angles
Consecutive angles of a parallelogram add up to `180^@`

Let `a` and `b` be consecutive angles (measured in degrees) such that the given relation holds:
`color(white)("XXX")a=9b-30`
and since `a+b=180 rarr b=180-a`
`color(white)("XXX")a=9(180-a)-30 = 1620-9a-30=1590-9a`
`color(white)("XXX")10a=1590`
`color(white)("XXX")a=159`
and
`color(white)("XXX")b=180-a=21`

Case 2: The given relation applies to opposite angles
Opposite angles of a parallelogram are equal.

Let `a` be the measure (in degrees) of the referenced opposite angles (and `b` be the measure of the other two angles: `a+b=180`)

We are told
`color(white)("XXX")a=9a-30`
Therefore
`color(white)("XXX")-8a=-30`

`color(white)("XXX")a=3 3/4`

`color(white)("XXX")b=180-a=176 1/4`

Answer 2

The angles are `21°, 159°, 21°, 159°`

Explanation:

In a parallelogram, there are only two sizes of angle - two are acute and two are obtuse.
Opposite angles are equal .
Consecutive angles are supplementary because they are co-interior angle on parallel lines.

An additional solution to the one given by Alan P is using the fact that the sum of the interior angles of a parallelogram is 360°.

Let one angle be `x`

The size of the other angle is given by `9x-30`

`2x+2(9x-30) = 360" "larr` sum of the angles is 360°

`2x +18x-60 = 360`

`20x =420`

`x = 21°`

The obtuse angles are each `180-21 = 159°`

The other option where the given angle is the opposite angle is described is well explained by Alan P.