You need to use the exterior angles. The sum of the exterior angles of any polygon is 360 deg. When the interior angle is 150 deg, the exterior angle is 180-150 = 30 deg, and when the interior angle is 170, the exterior is 180-170 = 10 deg. So

you have n/2 angles at 30, and n/2 at 10, so,

30n/2 + 10n/2 = 360, so,

20n =360, so,

n = 18.

2] the diagonals of a rhombus bisect each other at right angles, and they also bisect the angles through which they pass. So, if the point of intersection of the diagonals is E, then EB = 10 and AE = 6, and tan ABE = 6/10, = 0.6. Find ABE, then double it to find angle ABC. Hope this helps, Twiggy.

1.

The sum of the interior angles is:

(n/2)(150 + 170) = 160n deg.

The interior angles of a polygon always total 180(n – 2) deg.

Therefore:

180(n – 2) = 160n

180n – 360 = 160n

20n = 360

n = 18.

The polygon has 18 sides.

2.

The diagonals of a rhombus bisect each other at right angles.

The diagonals also bisect the angles of the rhombus through which they pass.

If O is the point where the diagonals meet, then:

AO / OB = 6 / 10 = 3/5.

Therefore ABO = arctan(3/5), and as OB bisects ABC,

ABC = 2arctan(3/5).

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