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°