Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 17: Geometry at Work 235

Of course, that assumes that someone told you that the triangles were congruent. What if they

asked you if the triangles were congruent? Technically, you’d have to measure all the sides and

all the angles of both triangles and check to see that you can match up parts of the first triangle

with parts of the second so that corresponding parts are congruent. That’s what the definition of

congruent triangles says.

That’s a lot of measuring, and you might realize that, for example, you don’t really have to mea-

sure the third angle of each triangle. If you measure the first two angles of both triangles and they

match, the third angles will have to match as well, because the three angles of any triangle always

add to 180r. That might get you to thinking about whether there’s anything else you can skip.

It turns out that to be certain that two triangles are congruent, you only need to check three

measurements from each triangle. You must always have at least one pair of congruent sides, to

guarantee that the triangles are the same size. If you only have the measurements of one side of

each triangle, and they are congruent, then you must have two pairs of matching angles to be

certain the triangles are congruent. It can be any two pairs, because if you know that two pairs of

angles are congruent, the third pair will match as well. If you measure two pairs of sides and find

that they match, and you can show that the angles they form also match, you can be certain the

triangles are congruent.

If you were to cut three sticks to set lengths and tried to put them together into triangles, you’d

find out there’s only one triangle you can make. If you measure all three pairs of sides in two

triangles and they match, you don’t need to measure any angles before you decide the triangles

are congruent. Three pairs of congruent sides congruent is enough.

Each of these “minimum requirements” involves three pieces, and each one has a three-letter

abbreviation.

B

C

A

S

R T

Minimums Required To Be Certain Triangles

Are Congruent

Minimum Needed Abbreviation

Example from

'ABC # 'STR

Three pairs of corresponding sides

congruent

SSS AB = ST

BC = RT

CA = RS

Two pairs of corresponding sides

and the angle between them

congruent

SAS AB = ST

BC = RT

’B = ’T

Two pairs of corresponding

angles and the side between them

congruent

ASA ’B = ’T

’C = ’R

BC = TR

Two pairs of corresponding angles

and a pair of sides not between them

congruent

AAS ’B = ’T

’C = ’R

AB = ST