The Art and Craft of Problem Solving

262 CHAPTER 8 GEOMETRY FOR AMERICANS

Now we search for equal angles. Since the diagonal BC is a transversal intersecting

parallel lines AB and CD, we have

LDCB= LABC.

Likewise,

LACB = LDBC,

since AC and BD are also parallel. Finally, we use the trivial observation that BC = BC

to use ASA to conclude that

MCB � l:::.DBC.

Consequently, the opposite sides have equal length and the opposite angles LA and

LD have equal measure. To show that the other pair of opposite angles are equal , we

need to draw diagonal AD and show that MCD � l:::.DBA.

To show that adjacent angles are supplementary, we either appeal to Fact 8.2.6

(AC is a transversal of the parallel lines AB and CD), or use the fact that the sum of the

angles in a quadrilateral is 3600 (Example 2.3.5 on page 45).

Finally, once the second diagonal AD is drawn, observing vertical angles and us-

ing congruence easily yields (c). •

This example used perhaps the most productive technique: drawing an auxiliary

object. Sometimes, as above, drawing a single line is enough to get the job done. This

method is powerful, but not always easy. It takes skill, experience, and luck to find the

right objects to add to a given figure. And if you add too many objects, your figure

becomes hopelessly confused. Always let the penultimate step be your guide, and keep

in mind what "natural" entities will help you. Triangles? Circles? Symmetry?

Here's an easy exercise.

8.2.10 A triangle is isosceles if two of its sides have equal length. Prove that if a

triangle is isosceles, the angles opposite the equal sides have equal measure.

8.2. 11 Let ABC be isosceles, with AB = AC. We call A the vertex angle of the isosce­

les triangle, and Band C are called the base angles. A nearly trivial consequence of

Fact 8.2.10 is

B =C = 90 - A/2;

in other words, the base angles of an isosceles triangle are complementary to half

the vertex angle (two angles are called complementary if they sum to a right angle;

contrast this with supplementary angles.)