CHAPTER 1

CONGRUENCY

1.1 Introduction

Assumed Knowledge

This text assumes a bit of knowledge on the part of the reader. For example, it assumes that you know that the sum of the angles of a triangle in the plane is 180° (x + y + z = 180° in the figure below), and that in a right triangle with hypotenuse c and sides a and b, the Pythagorean relation holds: c2 = a2 + b2.

We use the word line to mean straight line, and we assume that you know that two lines either do not intersect, intersect at exactly one point, or completely coincide. Two lines that do not intersect are said to be parallel.

We also assume certain knowledge about parallel lines, namely, that you have seen some form of the parallel axiom:

Given a line l and a point P in the plane, there is exactly one line through P parallel to l.

The preceding version of the parallel axiom is often called Playfair’s Axiom. You may even know something equivalent to it that is close to the original version of the parallel postulate:

Given two lines l and m, and a third line t cutting both l and m and forming angles ϕ and θ on the same side of t, if ϕ + θ < 180°, then l and m meet at a point on the same side oft as the angles.

The subject of this part of the text is Euclidean geometry, and the above-mentioned parallel postulate characterizes Euclidean geometry. Although the postulate may seem to be obvious, there are perfectly good geometries in which it does not hold.

We also assume that you know certain facts about areas. A parallelogram is a quadrilateral (figure with four sides) such that the opposite sides are parallel.

The area of a parallelogram with base b and height h is b · h, and the area of a triangle with base b and height h is b · h/2.

Notation and Terminology

Throughout this text, we use uppercase Latin letters to denote points and lowercase Latin letters to denote lines and rays. Given two points A and B, there is one and only one line through A and B. A ray is a half-line, and the notation denotes the ray starting at A and passing through B. It consists of the points A and B, all points between A and B, and all points X on the line such that B is between A and X.

Given rays and , we denote by ∠BAC the angle formed by the two rays (the shaded region in the following figure). When no confusion can arise, we sometimes use ∠A instead of ∠BAC. We also use lowercase letters, either Greek or Latin, to denote angles.

When two rays form an angle other than 180°, there are actually two angles to talk about: the smaller angle (sometimes called the interior angle) and the larger angle (called the reflex angle). When we refer to ∠BAC, we always mean the nonreflex angle.

Note. The angles that we are talking about here are undirected angles; that is, they do not have negative values, and so can range in magnitude from 0° to 360°. Some people prefer to use m(∠A) for the measure of the angle A; however, we will use the same notation for both the angle and the measure of the angle.

When we refer to a quadrilateral as ABCD we mean one whose edges are AB, BC, CD, and DA, Thus, the quadrilateral ABCD and the quadrilateral ABDC are quite different.

There are three classifications of quadrilaterals: convex, simple, and nonsimple, as shown in the following diagram.

1.2 Congruent Figures

Two figures that have exactly the same shape and exactly the same size are said to be congruent. More explicitly:

Theorem 1.2.1. Vertically opposite angles are congruent.

Proof. We want to show that a = b. We have

and it follows from this that a = b.

Notation. The symbol ≡ denotes congruence. We use the notation ΔABC to denote a triangle with vertices A, B, and C, and we use C(P, r) to denote a circle with center P and radius r.

Thus, C(P, r) ≡ C(Q, s) if and only if r = s.

We will be mostly concerned with the notion of congruent triangles, and we mention that in the definition, ΔABC ≡ ΔDEF if and only if the following six conditions hold:

Note that the two statements ΔABC ≡ ΔDEF and ΔABC ≡ ΔEFD are not the same!

The Basic Congruency Conditions

According to the definition of congruency, two triangles are congruent if and only if six different parts of one are congruent to the six corresponding parts of the other. Do we really need to check all six items? The answer is no.

If you give three straight sticks to one person and three identical sticks to another and ask both to constuct a triangle with the sticks as the sides, you would expect the two triangles to be exactly the same. In other words, you would expect that it is possible to verify congruency by checking that the three corresponding sides are congruent. Indeed this is the case, and, in fact, there are several ways to verify congruency without checking all six conditions.

The three congruency conditions that are used most often are the Side-Angle-Side (SAS) condition, the Side-Side-Side (SSS) condition, and the Angle-Side-Angle (ASA) condition.

Axiom 1.2.2. (SAS Congruency)

Two triangles are congruent if two sides and the included angle of one are congruent to two sides and the included angle of the other.

Theorem 1.2.3. (SSS Congruency)

Two triangles are congruent if the three sides of one are congruent to the corresponding three sides of the other.

Theorem 1.2.4. (ASA Congruency)

Two triangles are congruent if two angles and the included side of one are congruent to two angles and the included side of the other.

You will note that the SAS condition is an axiom, and the other two are stated as theorems. We will not prove the theorems but will freely use all three conditions.

Any one of the three conditions could be used as an axiom with the other two then derived as theorems. In case you are wondering why the SAS condition is preferred as the basic axiom rather than the SSS condition, it is because it is always possible to construct a triangle given two sides and the included angle, whereas it is not always possible to construct a triangle given three sides (consider sides of length 3, 1, and 1).

Axiom 1.2.5. (The Triangle Inequality)

The sum of the lengths of two sides of a triangle is always greater than the length of the remaining side.

The congruency conditions are useful because they allow us to conclude that certain parts of two triangles are congruent by determining that certain other parts are congruent.

Here is how congruency may be used to prove two well-known theorems about isosceles triangles. (An isosceles triangle is one that has two equal sides.)

Theorem 1.2.6. (The Isosceles Triangle Theorem)

In an isosceles triangle, the angles opposite the equal sides are equal.

Proof. Let us suppose that the triangle is ABC with AB = AC.

In ΔABC and ΔACB we have

so ΔABC ≡ ΔACB by SAS.

Since the triangles are congruent, it follows that all corresponding parts are congruent, so ∠B of ΔABC must be congruent to ∠C of ΔACB.

Theorem 1.2.7. (Converse of the Isosceles Triangle Theorem)

If in ΔABC we have ∠B = ∠C, then AB = AC.

Proof. In ΔABC and ΔACB we have

so ΔABC ≡ ΔACB by ASA

Since ΔABC ≡ ΔACB it follows that AB = AC.

Perhaps now is a good time to explain what the converse of a statement is. Many statements in mathematics have the form

where and are assertions of some sort.

For example:

If ABCD is a square, then angles A, B, C, and D are all right angles.

Here, is the assertion “ABCD is a square,” and is the assertion “angles A, B, C, and D are all right...