Angles and Shape Properties

A line of symmetry turns one unknown into two identical ones, which is what makes these questions solvable. An isosceles triangle with a forty-degree apex has a hundred and forty left for the two base angles, so each is seventy. Backwards is easier still: base angles of sixty-five leave an apex of fifty, one subtraction and no halving. Halve only when the pair is what you are looking for.

These questions chain their facts. The triangle's angle sum gives the base angle, seventy; that base angle sits on a straight line, so the angle beside it is a hundred and eighty minus seventy, a hundred and ten. Write each angle onto the diagram as you find it, because the angle the question actually asks for is usually two facts away from the one you are given, never one.

Two more totals finish the set. Angles meeting at a point add to three hundred and sixty, so a hundred and thirty-five and ninety leave a hundred and thirty-five. And the four angles of any quadrilateral also add to three hundred and sixty, so a hundred and twenty, eighty and a hundred leave sixty. Count the corners to decide which total applies — three corners means a hundred and eighty, four means three hundred and sixty.

A pyramid on an n-sided base has n plus one faces, two n edges and n plus one vertices, so a hexagonal pyramid has seven faces, twelve edges and seven vertices — and if the question asks for all three added together, twenty-six. A prism follows a different rule: n plus two faces, three n edges and two n vertices, so a hexagonal prism has eight, eighteen and twelve. Decide which family the solid is in before using a rule.

Three decisions cover the chapter. Count the corners to pick the angle total. Use the line of symmetry when it turns one unknown into a pair. And name the base before counting a solid's faces, edges and vertices. Label the diagram as you go, because almost every item here needs two facts, not one. Quiz and exercises now.