Sides $\overline{AH}$ and $\overline{CD}$ of regular octagon $ABCDEFGH$ are extended to meet at point $P$. What is the degree measure of angle $P$? i know its not 90 tho.

Interior angles of a regular octagon all have the same measure of 135 degrees. So if angle BAH has measure 135, then angle BAP (which is supplementary to BAH) has measure 45. Similarly, BCP (congruent to BAH) is also of measure 45 degrees.

Next, since ABC has measure 135 degrees (congruent to BAH), so the corresponding external angle has measure 225 degrees.

Now, ABCP is a quadrilateral. For any quadrilateral, its interior angles sum to 360 degrees. We have three of these angles, so we can easily find the last, APC.

[tex]m\angle BAP+m\angle_{\text{ext}} ABC+m\angle BCP+m\angle APC=360^\circ[/tex]

[tex]45^\circ+225^\circ+45^\circ+m\angle APC=360^\circ[/tex]

[tex]\implies m\angle APC=45^\circ[/tex]