The Art and Craft of Problem Solving

8.2 SURVIVAL GEOMETRY I 261

There is one critical fact about parallel lines that leads to many interesting lemmas.^2

Fact 8.2.6

(a) Alternate interior angles cut by a transversal are equal. In other words,

LBCD = LABC in the figure above. Notice that this equality is indicated in

the figure by the arcs that are drawn in.

(b) The converse of (a) is also true; i.e., if two lines are cut by a third, and the

alternate interior angles are equal, then these two lines are parallel.

Here are some corollaries of Fact 8.2.6.

8.2.7 The sum of the (interior) angles of a triangle equals 180°. Hint: Let ABC be a

triangle. Consider line BC (not the line segment), and construct a line parallel to BC

that passes through point A.

8.2.8 For any triangle, the measure of an exterior angle is equal to the sum of the two
other interior angles.

This example below is quite simple, but we are showing a fairly complete solution in
order to review basic problem-solving ideas and introduce new techniques that can be
used in much more elaborate ways later.

Example 8.2.9 A parallelogram is a quadrilateral whose opposite edges are parallel.

Prove that

(a) The opposite edges of a parallelogram have equal length.

(b) The opposite angles of a parallelogram are equal, and adjacent angles are

supplementary (add to 180°).

(c) The diagonals of a parallelogram bisect each other.

Solution: For (a), we need to show that AB = CD and AC = BD. What is the

penultimate step that allows us to conclude two lengths are equal? Our only tools so
far are congruent triangles. Hence we must force some triangles into existence. To do
so, we draw in a diagonal.

(^2) Parallel lines are a very rich and interesting topic, and we are deliberately simplifying the story in favor of a
strict "Euclidean" point of view. It turns out that Fact 8.2.6 is actually dependent on a postulate that is independent
of Euclid's other postulates. Modifying this postulate leads to different yet mathematically consistent alternatives
to Euclidean geometry, the so-called non-Euclidean geometries. See [19), [41), and [^2 9) for more details.