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Subsections

2 Triangles and congruence of triangles

2.1 Basic measurements

Three distinct lines, , and , no two of which are parallel, form a triangle. That is, they divide the plane into some number of regions; exactly one of them, the triangle, is bounded, and has segments of all three lines on its boundary.

The triangle with vertices is denoted by , where is the point of intersection of the lines and ; is the point of intersection of the lines and ; and is the point of intersection of the lines and . These points of intersection divide each of the lines into two unbounded half-lines and one bounded line segment, called a side of the triangle.

The triangle, , defines numbers, the angle measures (also called the angles) at the vertices , and , and the lengths of the sides, which are the line segments , and .

The angle measure at for example the vertex is denoted by , or .

2.2 Historical note

The use of the phrase measure of an angle'' is relatively modern. Up to about 50 years ago, the measure of the angle at was simply denoted by or , and it was left to the reader to distinguish between the angle and its measure. When convenient, we will follow this convention, and use the same notation for an angle and its measure.

2.3 More on measurements

We will always give angle measures in radians, so, if and all lie on a line, with between and , then .

We denote the length of the side , for example, by . Until modern times, the side and its length were denoted by the same symbol, and the reader had to figure out which is which from the context. As with angles, when convenient, we will also use the same notation for a line segment and its length.

The pair of lines, and , for example, determines two angles; the question of which of these angles is determined by the triangle can be stated in words with difficulty; we will leave this as visually obvious.

2.4 Congruence

Two triangles, and , are congruent if the corresponding angles have equal measures, and the corresponding sides have equal lengths. That is, the triangles, and are congruent if ; ; ; ; ; and . In this case, we write .

For physical triangles, two triangles are congruent if they exactly match if you put one on top of the other. Another way of saying this, for ideal triangles, is that there is an isometry of the plane (a composition of rotation, translation and reflection) that maps one exactly onto the other.

Exercise 2.1   Show that congruence of triangles is an equivalence relation.

It is essentially obvious that congruence of triangles is an equivalence relation. However, the statement that says nothing about whether is or is not congruent to . More precisely, the statement not only tells you that these two triangles are congruent, but also tells you that , , etc.

2.6 The axiom for congruence

Axiom 5 (ASA)   If , and , then .

It is common to refer to the above angle as Angle-Side-Angle'' or ASA.

For physical triangles this is essentially obvious. If you know the length of a side, and you know the two angles, then the lines on which the other sides lie are determined, so the third vertex is also determined.

2.7 Exercises

A physical triangle is determined by 6 pieces of information, the 3 lengths and the 3 angles. There are 6 possible statements concerning 3 pieces of information. Convince yourself that AAA and SSA are false, while AAS, ASA, SAS and SSS are true. (There is nothing here for you to hand in, but you need this information for the next two questions.)

Remark: One of these, AAS, is not obvious; in fact it is false in spherical geometry.

In the following few exercises, when you are asked to prove something you may assume that AAS, ASA, SAS and SSS are true. One other fact that you may use is Thm. 3.4: the sum of the angles of a triangle is . Note that this is only for these exercises; in general we cannot assume things we have not proven or taken as an axiom, because we may wind up applying circular reasoning (that is, giving a proof that something is true which implicitly assumes it was true to begin with.) But the main point of this exercise is to get you thinking about how geometry works, so we can relax our restrictions a little.

Exercise 2.2   Is it true that no 2 pieces of information suffice to determine a triangle? That is, can you find two pieces of information so that if you have any two triangles for which these two measurements are the same, the triangles must necessarily be congruent. Prove your answer.

Exercise 2.3   What about 4 pieces of information; i.e., do any four pieces of information suffice for congruence of triangles? Prove your answer.

A quadrilateral is a region bounded by four line segments; that is, it is a four-sided figure. The quadrilateral with vertices, , , and , in this order, is determined by the four line segments connecting and , and , and , and connecting and . For to form a quadrilateral, these segments must not intersect except at the verticies.

A quadrilateral defines 8 pieces of information: the lengths of the four sides, and the measures of the four angles. Two quadrilaterals are congruent if these 8 pieces of information agree.

What is the minimal number of pieces of information one needs about two quadrilaterals to prove that they are congruent? (No response needed here, but you need the answer for the next question.)

Exercise 2.4   State and prove one congruence theorem for quadrilaterals, where the hypothesis consists of the minimal number of pieces of information.

2.8 Monotonicity of lengths and angles

Here are two more axioms we shall need. Essentially, they say that for every real number, a segment can be scaled to that length, and that angles can be subdivided into angles of any measurement between 0 and .

Axiom 6 (Ruler Axiom)   If and are distinct points on a line, in that order, then . Further, for every positive real number , there is a point between and so that .

Axiom 7 (Protractor Axiom)   If is a line, and is a point on , then, for every number with , there is a line through so that the angles formed by and have measures and . Further, if , then there is a line passing through the sector of angle formed by and , so that and form an angle of measure .

Theorem 2.1 (SAS)   If and are such that , , and , then they are congruent.

Proof. Suppose we are given two triangles and as in the statement. If , then we would be done (by ASA).

So let us consider the case where they are different, and arrive at a contradiction. We may assume that (if not, just exchange the names on the triangles).

Apply the second part of Axiom 7 to find a line passing through the point and some point lying between and , so that .

By ASA, . Therefore . But we are given that . Therefore, . Since lies on the line determined by and , and lies between them, this contradicts Axiom 6.

2.9 Isosceles triangles

A triangle is isosceles if two of its sides have equal length. The two sides of equal length are called legs; the point where the two legs meet is called the apex of the triangle; the other two angles are called the base angles of the triangle; and the third side is called the base.

While an isosceles triangle is defined to be one with two sides of equal length, the next theorem tells us that is equivalent to having two angles of equal measure.

Theorem 2.2 (Base angles equal)   If is isosceles, with base , then .

Conversely, if has , then it is isosceles, with base .

Exercise 2.5   Prove Theorem 2.2 by showing that is congruent to its reflection . Note that there are two parts to the theorem, and so you need to give essentially two separate arguments.

2.10 Congruence via SSS

Theorem 2.3 (SSS)   If and are such that , and , then .

Proof. If the two triangles were not congruent, then one of the angles of would have measure different from the measure of the corresponding angle of . If necessary, relabel the triangles so that and are two corresponding angles which differ, with .

We find a point and construct the line so that , and . (That this can be done follows from Axioms 6 and 7.) It is unclear where the point lies: it could lie inside triangle ; it could lie on the line between and ; or it could lie on the other side of the line . We need to take up these three cases separately.

Exercise 2.6   Suppose the point lies on the line . Explain why this yields an immediate contradiction.

For both of the remaining cases, we draw the lines and . We observe that, by SAS, . It follows that and that . Hence is isosceles, with base , and is isosceles with base . Since the base angles of an isosceles triangle have equal measure, and .

First, we take up the case that lies outside ; that is, lies on the other side of the line from .

Exercise 2.7   Finish this case of the proof, first by showing that and . Then use the isosceles triangles to arrive at the contradiction that .

We now consider the case where lies inside . Extend the line to some point . Observe that , from which it follows that . Next, extend the line past to some point . Also extend the line past the point to some point , and extend the line past the point to some point .

Exercise 2.8   Finish this case of the proof by explaining why and , and then show that this leads to the contradiction .

2.11 Inequalities for general triangles

Theorem 2.4 (exterior angle inequality)   Consider the triangle . Let be some point on the line , where lies between and . Then
(i)
.
(ii)
.

Proof. We first prove part (i). Let be the midpoint of the line segment ; that is, lies on the line , between and , and . Draw the line and extend it past to the point , so that is the midpoint of . Also draw the line .

Exercise 2.9   Finish the proof of part (i). Hint: First show that (Thm. 1.1 may be useful.) Use that to compare and , and conclude that .

For part (ii), we choose to be the midpoint of the line , and extend beyond to , so that . Also, extend the line Now extend the line beyond to some point .

Exercise 2.10   Finish the proof of part (ii). First show that , and then compare , , , and .

The next theorem says that in a triangle, if one angle is bigger than another, the side opposite the bigger angle must be longer than the one opposite the smaller angle. This is generalizes the fact that the base angles of isosceles triangles are equal (Thm. 2.2).

Theorem 2.5   In , if , then we must have .

Proof. Assume not. Then either or .

Exercise 2.11   Show that if , the assumption is contradicted.

Exercise 2.12   Now assume , find the point on so that , and draw the line . Finish the proof in this case. Hint: Use Thm. 2.4 and the fact that to conclude that . Now observe that . Explain why this gives the contradiction .

The converse of the previous theorem is also true: opposite a long side, there must be a big angle.

Theorem 2.6   In , if , then .

Proof. Assume not. If , then is isosceles, with apex at , so , which contradicts our assumption.

If , then, by the previous theorem, , which again contradicts our assumption.

The following theorem doesn't quite say that a straight line is the shortest distance between two points, but it says something along these lines. This result is used throughout much of mathematics, and is referred to as the triangle inequality''.

Theorem 2.7 (the triangle inequality)   In , we have

.

Exercise 2.13   Prove the triangle inequality: First extend to a point so that , then form the isosceles triangle . Use this triangle and Thm 2.2 to show that . Conclude that by using another theorem from this section. Then show that .

2.12 Congruence via AAS

Theorem 2.8 (AAS)   Suppose we are given triangles and , where , , and . Then .

Proof. We first observe that, by either SAS or ASA, if , then . Hence we can assume that , from which it follows that either or . We can assume without loss of generality that (that is, if we had that , then we would interchange the labelling of the two triangles).

Now find the point between and , so that . Observe that, by SAS, . Hence . This contradicts that fact that, since is an exterior angle for , we must have that .

This concludes the generalities concerning congruence of triangles. We now know the four congruence theorems, ASA, SAS, SSS and AAS. We also know that the other two possibilities, SSA and AAA, are not valid. It follows that, for example, if we are given the lengths of all three sides of a triangle, then the measures of all three angles are determined. However, we do not as yet have any means of computing the measures of these angles in terms of the lengths of the sides.

2.13 Perpendicularity and orthogonality

Two lines intersecting at a point are perpendicular or orthogonal if all four angles at are equal. In this case, each of the angles has measure . These angles are called right angles. It is standard in mathematics to use the words perpendicular and orthogonal interchangeably.

BASIC CONSTRUCTION. Given a line , and any point , there is a line through perpendicular to .

Exercise 2.14   Prove that the line through perpendicular to is unique. (Note that may or may not lie on .)

In any triangle, there are three special lines from each vertex. In , the altitude from is perpendicular to ; the median from bisects (that is, it crosses at a point so that ); and the angle bisector bisects (that is, if is the point where the angle bisector meets , then ).

Theorem 2.9   If is the apex of the isosceles triangle , and is the altitude, then is also the median, and is also the angle bisector, from .

Exercise 2.15   Prove this theorem. (Hint: Construct the altitude and apply AAS to the pair of resulting triangles.)

Theorem 2.10   In an isosceles triangle, the three altitudes meet at a point.

Proof. Let be the apex of the isosceles , and let be the altitude, which is also the median and the angle bisector. Similarly, let be the endpoint on of the altitude from , and let be the endpoint on of the altitude from . Let be the point of intersection of with , and let be the point of intersection of with . We need to prove that .

By AAS, . Hence . Since is also the angle bisector, by ASA, . Hence , from which it follows that .

Exercise 2.16   Prove that the three angle bisectors in an isosceles triangle meet at a point.

Exercise 2.17   Prove that the three medians in an isosceles triangle meet at a point.

Next: 3 The parallel axiom Up: MAT 200 Course Notes on Previous: 1 Introduction
Scott Sutherland 2002-12-18