Question

Four grey polygons join together as shown in the diagram. I’ve calculated angle a but I don’t know angle b. Angle a is 108

What is b? Thanks

Answer 1

Part 1)
`a=108^o`

Part 2)
`b=24^o`

Explanation:

Part 1) Find the measure of angle a

we know that

The formula to calculate the measure of the interior angle of a regular polygon is equal to

`(180^o(n-2))/(n)`

where

n is the number of sides of the regular polygon

For the regular pentagon in the figure

n=5

substitute in the formula

`a=(180^o(5-2))/(5)`

`a=(180^o(3))/(5)`

`a=108^o`

Part 2) Find the measure of angle b

Remember that a regular polygon has equal sides and equal interior angles

so

In the figure , the length of the side of the two smaller equilateral triangles is equal to the length of the side of the regular pentagon

therefore

The triangle in white of the figure is an isosceles triangle

The measure of the vertex angle of the isosceles triangle is equal to

`360^o-(108^o+60^o+60^o)=132^o`

Remember that

The sum of the interior angles of triangle must be equal to 180 degrees

so

`2b+132^o=180^o`

solve for b

`2b=180^o-132^o`

`2b=48^o`

`b=24^o`