Actually, I don't have much to say about today's geometry lesson at all, since Lesson 8-5 of the U of Chicago text is on the areas of triangles. You already know that the area of a triangle is half the product of its base and height. I did point out that this is proved for all three cases -- when the altitude is inside, outside, or aside of the triangle. This roughly corresponds to the acute, obtuse, and right triangles. But notice that if the altitude is drawn from the largest angle of a triangle, the altitude is always inside the triangle even if that angle is right or obtuse.
One of the included questions gives the area of a quadrilateral with perpendicular diagonals as half the product of those diagonals. It applies to all quadrilaterals whose diagonals are perpendicular -- which includes kites (and rhombuses and squares, which are kites under the inclusive definition). In some texts this is a highlighted formula, but in the U of Chicago it is buried in an exercise.