Question 280329
ABCDEFGH is a regular octagon.


If so, then each interior angle is the same.


Each angle of an octagon can be found using the following general formula:


i = ((n-2)*180)/n where i = each interior angle and n = number of sides.


In a triangle, n = 3, and each interior angle would be 1*180/3 = 60 degrees.


In a rectangle, n = 4, and each interior angle would be 2*180/4 = 90 degrees.


In an octagon, n = 8, and each interior angle would be 6*180/8 = 1080/8 = 135 degrees.


Now your octagon is labeled ABCDEFGH.


You want to find the measure of:


ABC
ACD
ABD


Angle ABC is one of the interior angles so angle ABC equals 135 degrees.


A picture of angle ABC is shown below:


<img src = "http://theo.x10hosting.com/problems/280329p1.jpg" alt = "********** PICTURE DID NOT DISPLAY PROPERLY **********" />


Angle ACD = 112.5 degrees as shown in the following picture.


<img src = "http://theo.x10hosting.com/problems/280329p2.jpg" alt = "********** PICTURE DID NOT DISPLAY PROPERLY **********" />


As you can see, Angle ACD creates a small triangle ABC.   Since this is a regular polygon, then triangle ABC is an isosceles triangle.   Since angle ABC is 135 degrees, the other 2 angles of the triangle have to be (180-135)/2 which equals 22.5 degrees.   Since angle BCD is also equal to 135 degrees, and angle BCA is equal to 22.5 degrees, then angle ACD is equal to 135 minus 22.5 = 112.5 degrees.


Angle ABD = 112.5 degrees as shown in the following picture.


<img src = "http://theo.x10hosting.com/problems/280329p3.jpg" alt = "********** PICTURE DID NOT DISPLAY PROPERLY **********" />


As you can see, Angle ABD creates a small triangle BCD.   Since this is a regular polygon, then triangle BCD is an isosceles triangle.   Since angle BCD is 135 degrees, the other 2 angles of the triangle have to be (180-135)/2 which equals 22.5 degrees.   Since angle ABC is also equal to 135 degrees, and angle CBD is equal to 22.5 degrees, then angle ABD is equal to 135 minus 22.5 = 112.5 degrees.