Question

Find the measures of the angles formed by​ (a) two consecutive radii and​ (b) a radius and a side of a regular dodecagon.

Answer 1

see the picture to better understand the problem

we know that
A regular dodecagon has 12 sides
Central angle of a regular polygon is an angle whose vertex is the center of the polygon with two consecutive radii
thus 
the measure of a central angle is
m∠AOC=360/n
where n is the number of sides
so
m∠AOC=360/12-----> 30°

therefore

the answer Part a) is 
the measure of the angle formed by two consecutive radii is 30 degrees

Part b) 
we know that
OAC is an isosceles triangle
so
OA=OC
m∠OAC=m∠OCA

the sum of the internal angles of a triangle is 180 degrees
so
180=2*m∠OAC+m∠AOC-----> m∠OAC=[180-30]/2----> 75°

the answer part b) is 
the measure of the angle formed by a radius and a side is 75°

Answer 2

A regular dodecagon is a regular polygon, and it as all its sides to be congruent.

(a) The angle formed by two consecutive radii

• A regular dodecagon has 2 congruent sides.
• The measure of angle at the center of the regular dodecagon is 360 degrees.

So, the angle between consecutive radii is:

$\theta = \frac{360}{12}$

$\theta = 30$

Hence, the angle formed by two consecutive radii is 30 degrees

(b) The angle formed by a radius and the side of a regular dodecagon

The triangle in one part of a regular dodecagon is a isosceles triangle.

The required angle is the base angle of the triangle.

And it is calculated using:

$2 \alpha + \theta = 180$

This gives

$2 \alpha + 30= 180$

Subtract 30 from both sides

$2 \alpha = 150$

Divide both sides by 2

$\alpha = 75$

Hence, the angle formed by a radius and the side of a regular dodecagon is 75 degrees

Read more about regular polygons at:

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