# point P is in a regular octagon ABCDEFGH so that triangle ABP is equilateral. how many degrees in angle APC?

Relevance

The angle APB is 60 degrees and the triangle is equilateral, therefore angle ABP must be 60 degrees. By definition, the angle ABC must be 135 degrees for an octagon, so the angle CBP must be 75 degrees.

Points C, B, and P form an isosceles triangle with angle CBP equal to 75 degrees. By definition, angles BCP and CPB must be equal to half of (180-CBP), or 52.5 degrees.

Here is a diagram:

http://www.squidzone.ca/UT/Picture%200006.jpg

Angle APC is the sum of angles APB and CPB, or 60+52.5 or 112.5 degrees.

• if it is regular octagon, APB can't be equilateral (for this all three angles would have to be 60deg) - it is isosceles.

360deg/8=45deg which is APB angle (from A to B) in a regular octagon.

APC is just double angle (from A to C) of that of APB. result is 90deg.

• m<ABC = 180º - 360º/8 = 135º

m<PBC = 135º - 60º = 75º

m<APB = m<ABP = m<BAP = 60º

Let s = AB ...... one side of octagon

AP = BP = BC = s

m<BPC = m<BCP ................ because BP = BC

m<BPC + m<BCP + m<PBC = 180º......... triangle BPC

m<BPC + m<BCP + 75 = 180º

m<BPC + m<BCP = 105º

2m<BPC = 105º

m<BPC = 52.5º

m<APC = m<APB + m<BPC = 60º + 52.5º = 112.5º