The Art and Craft of Problem Solving

8.2 SURVIVAL GEOMETRY I 259

Please do not read passively. The more of these simple problems that you do now,

the better off you will be when confronted with genuinely tricky questions.

We call Sections 8.2-8.3 "survival geometry" because they contain a lean but ade­

quate stock of facts and techniques that will allow you to get started on most problems.
If you master the facts and lemmas and fearlessly employ the problem-solving ideas
presented in the next two sections, you will be able to tackle an impressive variety of
challenging questions.

Points, Lines, Angles, and Triangles

We assume that you know, at least intuitively, the meaning of points, lines, angles,
triangles, and polygons, and that you know that parallel lines are "lines that never
meet." A straight angle is the angle made by three points on a line; i.e., an angle with

magnitude n or 180° (we will use radians and degrees interchangeably, depending on

convenience). A right angle is, of course, half of a straight angle, or 90°; four right

angles are produced when two perpendicular lines meet.

Lines are understood to extend infinitely in both directions. If a line extends only

in one direction, it is a ray, if it begins and ends at two points, it is a line segment.

Using precise notation, the symbols

AB, AB, AB, AB

denote, respectively, the line passing through A and B, the ray starting at A and passing

through B, the line segment between A and B, and the length of this line segment.
However, we may informally write "line AB" or just "AB," and it should be obvious
by context whether we refer to a line, a line segment, or a length.
Likewise, three points A, B, C, taken in order, determine an angle. The angle itself
is denoted by LABC, but its measure is more precisely denoted by mLABC. For ex­

ample, we may write mLABC = 80°. Informally, we will write LABC to denote either

the angle or its measure , and sometimes just write ABC; again, it should always be
clear from context. When a figure is simple and uncluttered, we may even get away
with referring to an angle by a single point. For example, in the figure below, LA is
unambiguous, but L.C is not. The interior angle of MBC at Cis L.ACB, while the
exterior angle is LACD.

A

G

B C 'D

Two triangles are congruent if it is possible to move one and superimpose it exactly
over the other, possibly after rotating it in space. Congruent triangles are the "same" in
that all corresponding angles and lengths are equal. Using precise notation, we would
write
MBC� l:::.DEF,