Step-by-step explanation: We have two polygons, one with five sides and the other with eight sides. The question states that the pentagon has exactly one line of symmetry which means the line that runs down from point D to line AB divides the shape into exactly two equal sides. Hence angle A measures the same size as angle B (in the pentagon).

First step is to calculate the angles in the pentagon. The sum of angles in a polygon is given as

(n - 2) x 180 {where n is the number of sides}

= 3 x 180

= 540

This means the total angles in the pentagon can be expressed as

A + B + 84 + 112 + 112 = 540

A + B + 308 = 540

Subtract 308 from both sides of the equation

A + B = 232

Since we have earlier determined that angle A measures the same size as angle B, we simply divide 232 into two equal sides, so 232/2 = 116

Having determined angle A as 116 degrees, we can now compute the value of angle A in the octagon ABFGHIJK. Since the figure is a regular octagon, that means all the angles are of equal measurement. So, the sum of interior angles is given as

(n - 2) x 180 {where n is the number of sides}

= 6 x 180

= 1080

If the total sum of the interior angles equals 1080, then each angle becomes

1080/8

= 135 degrees.

That means angle A in the octagon measures 135, while in the pentagon it measures 116. The size of angle x is simply the difference between both values which is

x = 135 - 116

x = 19 degrees

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3 years ago