My 6th-grader, Lydia, did not exhibit much interest in the pictured "exceeding standard question" about the regular octagon. She's cool with "meets requirements," especially when her friends are facetiming her. You can see the blank space for work and the perfunctory line drawn between points H and F, which was supposed to persuade me that she'd made an effort. Here are the steps of the approach I described to her while she glanced back and forth between the paper and her dinging phone.

1. You know the interior angles of a triangle sum to 180 degrees. It goes up by 180 degrees for every side added: for a quadrilateral (such as a rectangle) the interior angles sum to 360 degrees, for a pentagon 540 degrees, etc. A formula is: Sum of Interior Angles = 180(# of sides - 2).

2. So the interior angles of an octagon add up to 180 x 6 = 1080 degrees. Since it's a regular octagon (all sides equal length), the eight angles all have the same measure of 1080/8 = 135 degrees.

3. You're supposed to find the measure of ∠HAF, which can be thought of as the sum of ∠HAG and ∠GAF. ∠HAG is 22.5 degrees (because ∠GHA is 135 degrees and ∠HAG is half the remaining 45 degrees of triangle AHG). For the same reason, ∠BAC is also 22.5 degrees.

4. Is this enough to persuade you that the 135 degrees of ∠HAB is made of six equal "little angles" all with vertex A? [Bored look.] Note that 22.5 x 6 = 135. [Completely impassive.]

5. If not, the hypothesis can be checked. Since the octagon is "regular," opposite sides are parallel and ∠AFE is 90 degrees. This means that ∠GFA is 45 degrees (since we know that ∠GFE is 135 degrees). Also, since ∠HGF is 135 degrees, and ∠HGA is 22.5 degrees, ∠AGF must be 135 - 22.5 = 112.5 degrees.

6. Therefore ∠GAF = 180 - 45 - 112.5 = 22.5 degrees. And ∠HAF = ∠HAG + ∠GAF = 22.5 + 22.5 = 45 degrees.

Lydia misses baseball, because in 5th grade, when the Twins were playing on TV, it was her homework, not mine.

## Comments